Metrical results on the geometry of best approximations for a linear form

Consider the integer best approximations of a linear form in $n\ge 2$ real variables. While it is well-known that any tail of this sequence always spans a lattice is sharp for any $n\ge 2$. In this paper, we determine the exact Hausdorff and packing dimension of the set where equality occurs, in terms of $n$. Moreover, independently we show that there exist real vectors whose best approximations lie in a union of two two-dimensional sublattices of $\Z^{n+1}$. Our lattices jointly span a lattice of dimension three only, thereby leading to an alternative constructive proof of Moshchevitin's result. We determine the packing dimension and up to a small error term $O(n^{-1})$ also the Hausdorff dimension of the according set. Our method combines a new construction for a linear form in two variables $n=2$ with a result by Moshchevitin to amplify them. We further employ the recent variatonal principle and some of its consequences, as well as estimates for Hausdorff and packing dimensions of Cartesian products and fibers. Our method permits much freedom for the induced classical exponents of approximation.


Best approximations in small sublattices
Let ξ = (ξ 1 , . . ., ξ n ) ∈ R n and for simplicity consider the maximum norm on R n denoted by . .For convenience we introduce the notation ξ * = (ξ, 1) ∈ R n+1 .A classical topic in Diophantine approximation is to study small absolute values of a linear form (1) q • ξ * = q 1 ξ 1 + • • • + q n ξ n + q n+1 , for non-zero integer vectors q = (q 1 , . . ., q n+1 ).Here small means compared to q , where x ∈ R n denotes the restriction of x ∈ R n+1 to its first n coordinates.We assume (1) does not vanish for any integer vector q = 0, and then call ξ totally irrational.We call a vector q ∈ Z n+1 a best approximation for ξ if where the minimum is taken over all b ∈ Z n+1 with norm of b = 0 at most q . 1 They are unique up to sign for totally irrational ξ.Considering the set of all best approximations Middle East Technical University, Northern Cyprus Campus, Kalkanli, Güzelyurt johannes@metu.edu.tr ; jschleischitz@outlook.com. 1 We follow the classical definition as for example in [14], that indeed uses the norms of the restricted integer vectors without last coordinate (constant term).When omitting the "hat", even though q ≍ ξ q , the concrete sequence may change in some cases.Our results are valid with either definition.
with norms in increasing order gives rise to the sequence q j ∈ Z n+1 , j ≥ 1, of best approximations associated to ξ with the properties Let us adapt the notation R(ξ) from [14] to denote the minimum integer R so that some tail of the best approximations (q j ) j≥j 0 lies in an R-dimensional sublattice L = L(ξ) of Z n+1 that may depend on ξ.It is well-known that for n ≥ 2 and totally irrational ξ ∈ R n , we have R(ξ) ≥ 3. See [13,Theorem 1.2] for the claim with its proof sketched in the same paper [13, § 1.3], and [14,Theorem 7] for generalizations to a system of linear forms.On the other hand, Moshchevitin [13] showed that the following sets are not empty.
Definition 1.For n ≥ 2 an integer, let Γ n be the set of all totally irrational ξ ∈ R n inducing R(ξ) = 3.
By the above observation, the set R 2 \ Γ 2 is a countable union of rational affine hyperplanes, hence we may assume n ≥ 3.More precisely, denoting by dim H and dim P the Hausdorff and packing dimension respectively, Moshchevitin's refinements [14, Theorems 12 & 13] of his own result directly imply the following fact.
Theorem 1.1 (Moshchevitin).We have dim In our first new result, relying on auxiliary results from [14] and [4,5], we determine the exact Hausdorff and packing dimension of the sets Γ n .
Theorem 1.2.The Hausdorff dimensions of Γ n are given as (2) dim and their packing dimensions as The lower bounds follow relatively easily from combining observations from [4,5,14], together with some metrical theory of Cartesian products.The upper bounds require more work, especially the three-dimensional sublattice in the definition of Γ n being arbitrary causes our proof to become technical.Define the uniform exponent of approximation with respect to a linear form as and the ordinary exponent of approximation Then by Dirichlet's Theorem for any ξ ∈ R n we have (4) ω(ξ) ≥ ω(ξ) ≥ n.
Generalizing the proof strategy of Theorem 1.2, we can obtain similar results on best approximations in sublattices of higher dimension k.It turns out that the packing dimension of the accordingly defined sets does not increase up to k = n.
The main focus of this paper is to study the following problem on best approximations for a linear form.
Problem 1.For n ≥ 2, does there exist totally irrational ξ ∈ R n so that some tail of best approximations (q j ) j≥j 0 lies in a finite union of two-dimensional sublattices of Z n+1 ?If so, determine the minimum possible number N = N(n) of sublattices.Determine/Estimate the Hausdorff and packing dimensions of these sets as functions of n.
Clearly N ≥ 2 for n ≥ 2 by the fact R(ξ) ≥ 3 for totally irrational ξ recalled above.
Remark 1.For simultaneous approximation, the answer is negative, as any hyperplane contains only finitely many best approximations as soon as ξ is totally irrational.Moreover, in either linear form or simultaneous approximation problem, clearly any onedimensional subspace contains at most one best approximation vector for given ξ, if so its (up to sign) unique integer vector with coprime coefficients.
Remark 2. As pointed out to the author in private communication by N.G.Moshchevitin, with some effort a positive answer to Problem 1 (omitting the metrical aspects) with the optimal constant N = 2 can be derived from the lemma in [10] and its proof.This lemma essentially states the following: Let G ⊆ Z n be a set of integer vectors that is not contained in a set of the form ℓ ∪ F , where ℓ is a line and F a finite set, in R n .Then there is totally irrational ξ ∈ R n such that for integer vectors within the set G * = {(g, h) : g = (g 1 , . . ., g n ) ∈ G, h ∈ Z} ⊆ Z n+1 we find very small linear forms |q • ξ * | for certain q ∈ G * (implying q ∈ G), occurring with some density that in particular admits to ask for ω(ξ) = ∞.Taking G the union of two non-collinear rational one-dimensional sublattices (lines) ℓ 1 , ℓ 2 of Z n , its embedding G * in Z n+1 lies in the union of the two two-dimensional sublattices ℓ 1 , e n+1 Z and ℓ 2 , e n+1 Z of Z n+1 .As pointed out to the author by N.G.Moshchevitin, with some cumbersome additional geometrical arguments one can guarantee that these integer points indeed form some tail of the best approximations associated to ξ.However, this is not explicitly carried out in the short note [10] (nor in later work) as it was not of relevance in that paper.In our alternative construction regarding Problem 1 below (proof of Theorem 2.1), we fix the lines ℓ i = e i Z , i = 1, 2 as the first two coordinate axes of R n and provide an explicit, rather elementary argument of this fact for certain ξ, based on Minkowski's Second Convex Body Theorem.Moreover, we address the metrical problem.

On Problem 1
2.1.Main new results.As indicated in Remark 2, we give an almost complete answer to Problem 1. Indeed, we show that for arbitrary n and the maximum norm the answer is positive, and the optimal bound N(n) = 2 can be reached for any n ≥ 2.Moreover, we may choose the two two-dimensonal sublattices so that they span a lattice of dimension only three in Z n+1 .Thereby we recover Moshchevitin's result that R(ξ) = 3 can be reached, i.e.Γ n = ∅, with a new proof, that we consider easier than the original one from [13].
To state our result in full generality, we need to introduce some notation.Let us first define the two-dimensional sublattices of Z n+1 given by i.e. the Z-span of the i-th and the (n + 1)-st canonical base vector in R n+1 .For the immediate concern of Problem 1, we will only need H 1 and H 2 .Let us define the following properties of ξ ∈ R n with induced sequence of best approximations (q j ) j≥1 : (i) ξ is totally irrational (ii) Some tail (q j ) j≥j 0 lies in the union of the two-dimensional sublattices H 1 and H 2 of Z n+1 .(iii) Some tail (q j ) j≥j 0 lies in the three-dimensional sublattice (iv) Large best approximations lie in H 1 and H 2 alternatingly, i.e. for any j ≥ j 0 (v) For j ≥ j 0 , three consecutive best approximations q j , q j+1 , q j+2 are linearly independent, thus q j , q j+1 , q j+2 R ∩ Z n+1 has full dimension three in e 1 , e 2 , e n+1 Z (vi) For w ∈ [n, ∞], we have We comment on the conditions and their mutual relations in § 2.2 below.We want to remark that we expect Theorem 2.1 below to remain true when replacing e 1 in H 1 and e 2 in H 2 by any pair of linearly independent integer vectors in the two-dimensional subspace of R n+1 defined by x 3 = • • • = x n+1 = 0.This is supported by [10], see Remark 2 above.Definition 2. Let n ≥ 2 be an integer.Define Θ n ⊆ R n as the set of ξ ∈ R n for which conditions We can now finally state the following rather satisfactory partial answer to Problem 1.
In particular Θ n = ∅ and N = 2 can be reached in Problem 1. Conversely, we have In view of (8), estimates (12) are an obvious consequence of (2) from Theorem 1.2.The bound (11) can be improved with some effort, see Remark 5 below.On the other hand, we are unable to provide a non-trivial upper bound for dim P (Θ 2 ).Comparing the bounds of Theorem 1.2 and Theorem 2.1 yields the following corollary.
The following problem remains open.
An explicit lower bound slightly exceeding n−2+1/(n 2 +1) for the Hausdorff dimension of Θ n can be readily deduced from the proof of Theorem 2.1 below, see formula (62).Inserting small n in this strengthened bound (62), the decimal expansions start with (13) dim See also Remark 8 below on bounds for Hausdorff or packing dimension when additionally restricting ω(ξ), complementing the left estimate of (9).Our method suggests the following refinements of (9) (and (62)).
See Remark 7 in § 9.4 below for more details.For small n, Conjecture 1 would yield considerable improvements of (13).While the right inequality is again strict, asymptotically our results suggest just a small improvement, with lower bound still of order n − 2 + 2/n 2 − O(n −4 ).We wonder if the special choice of H 1 , H 2 in the sets Θ n (w), Θ n are significant.This is constituted in the following more general problem.Problem 3. Do the lower bounds of Theorem 2.1 (possibly Conjecture 1) hold for the set of ξ ∈ R n with the property that some tail of best approximations (q j ) j≥j 0 lies in a union of any two fixed two-dimensional sublattices of Z n+1 ?Do the upper bounds of Theorem 2.1 (possibly Conjecture 1) hold for the set of ξ ∈ R n with some tail (q j ) j≥j 0 in a union of any two (or finite?) two-dimensional sublattices of Z n+1 , independent of ξ?
It seems the method of § 5.3 can be used to verify the second part of Problem 3 for the special case of the two sublattices jointly spanning a three-dimensional lattice in Z n+1 .This may be a necessary and sufficient criterion for the lower bounds as well.
The main substance of Theorem 2.1 are the lower bounds.In short, to prove (9), we combine a new construction for n = 2 with a result by Moshchevitin [14,Theorem 12].It follows that a "generic" vector in R n−2 , in sense of Lebesgue measure, gives rise to vectors in Θ n (w) by adding two more suitable real components.This further directly implies the lower bound n − 2 for the Hausdorff dimension of the sets Θ n (w).With some refined argument and using metrical results by Sun [16] and by Das, Fishman, Simmons, Urbański [4,5] we find the stronger lower bounds in (9).The upper bound (11) not implied by Theorem 1.2 follows independently from a classical formula by Jarník [8] and the theory of continued fractions.
By small modifications of the proof of Theorem 2.1, we can obtain best approximations ultimately lying in a union of k two-dimensional sublattices of Z n+1 , that together span a (k + 1)-dimensional space, but in no smaller number of two-dimensional subspaces.We want to explicitly state this generalization of the case k = 2 of Theorem 2.1, but avoid detailed metrical formulas for brevity.Theorem 2.2.Let n ≥ 3.For any 2 ≤ k ≤ n, there exists a set of Hausdorff dimension strictly greater than n − k, consisting of ξ ∈ R n with property (i) and (ii * ) Some tail of the best approximation sequence (q j ) j≥j 0 lies in the union . ., e k , e n+1 Z ⊆ Z n+1 .(vii) No tail (q j ) j≥j 1 lies in union of less than k two-dimensional sublattices of Z n+1 , similarly no tail is contained in a sublattice of dimension k or less.
In fact analogues of all (i)-(vi) hold for the ξ in Theorem 2.2.For fixed n, it is natural to expect that the Hausdorff dimension of the set in Theorem 2.2 increases with k.This is not reflected in the claim.For k = n, the according set has full n-dimensional Lebesgue measure (a rigorous argument for this follows from similar method in § 5.2 below).An immediate corollary of Theorem 2.2 reads as follows.
Probably Corollary 2 could be derived independently from ( 6), (7) upon determining (estimating) the involved Hausdorff dimensions.For sake of completeness, we end this section with estimating the size of the set Y n,k of ξ ∈ R n inducing infinitely many best approximations in any finite union of k-dimensional sublattices L 1 , . . ., L i(ξ) , depending on ξ, of Z n+1 .The proof is not complicated.
Theorem 2.3.Let n ≥ k ≥ 2 be integers and Y n,k be as above.Then For n = k = 2 the upper bound becomes 5/3.One may compare this with the smaller bound 7/5 from (11) for the smaller set Θ 2 ⊆ Y 2,2 dealing with a special collection of two-dimensional sublattices in which all but finitely many best approximations lie.Problem 4. Are the packing dimensions of Y n,k full?What if we instead require all large best approximations to lie in L 1 , . . ., L i(ξ) ?2.2.On the conditions (i)-(vi).Clearly (iv) ⇒ (ii) ⇒ (iii) and (iv) ⇒ (i).Moreover (iv) ⇒ (v) by the observation on one-dimensional subspaces from Remark 1.So (i), (ii), (iii) and (v) are rather stated for sake of completeness.Moreover, the theory of continued fractions and Dirichlet's Theorem (4) easily imply that conversely (ii) ⇒ (iv), see the proof in § 8.So (ii) ⇔ (iv).We want to comment on (vi) in the light of Theorem 1.1.The subset of vectors within Γ n originally constructed by Moshchevitin in [13] have the property ω(ξ) = ∞.Thus they form a set of Hausdorff dimension at most n − 2 in view of [5,Theorem 3.6], so the metrical claim in Theorem 1.1 cannot be improved in this way.However, using the refinements from [14] together with some new idea for linear forms in two variables, we will find ξ satisfying (i)-(v) and with finite uniform exponent of approximation.This enables us to surpass this treshold value n − 2 for the Hausdorff dimension even for the smaller sets Θ n ⊆ Γ n in Theorem 2.1.

A structural result on Γ n
In the proof of Theorem 1.2, we show via [14,Theorem 12] the following: For any for the smaller set Γ n with special choice of the three-dimensional lattice e 1 , e 2 , e n+1 Z ).The sets F n (ξ 1 , ξ 2 ) are hereby implicitly derived from the convergence part of the Borel-Cantelli Lemma within the proof in [14].In our last new result, we provide an explicit set for F n independent of the choice of ξ 1 , ξ 2 in terms of Diophantine properties, upon increasing the lower bound on the uniform exponent to 3n − 4.
Theorem 3.1.Let n ≥ 2. For any vector ξ = (ξ 1 , . . ., ξ n ) in the set the tail of best approximations lies in the three-dimensional sublattice L n,3 = e 1 , e 2 , e n+1 Z of Z n+1 .In particular See also Remark 10 below for refinements.From Theorem 3.1 together with metrical results from [4,5], we get another proof for the lower bound n − 1 for the packing dimension of Γ n .For its Hausdorff dimensions, using a classical result of Khintchine [9] and again [4,5], we can deduce from Theorem 3.1 the lower bound n − 2 + 2/(3n − 4) for n ≥ 3, weaker than the bound in Theorem 1.2.In fact both claims again hold for Γ n .While there are some similarities underying the fundamental ideas of Theorem 3.1 and [14, Theorem 12], the proofs differ considerably.Theorem 3.1 uses Minkowski's Second Convex Body Theorem instead of the Borel Cantelli Lemma.
The choice of the maximum norm in our new results is just for convienence, we may establish the analogous result for a large class of norms, including all p-norms ( |x i | p ) 1/p , by small modifications of the proof.In fact, at least in Theorems 1.2, 3.1, we may take any norm.

4.
1.An auxiliary result by Moshchevitin.We recall a partial result of [14,Theorem 12].In the notation of [14], its special case n = 1, m = 2 and m * equal to our present n, yields: Theorem 4.1 (Moshchevitin).Let n ≥ 3.If q j = (q j,1 , q j,2 , q j,3 ) ∈ Z 3 is the sequence of best approximations for (ξ 1 , ξ 2 ) ∈ R 2 and we have then for almost all choices of remaining entries (ξ 3 , . . ., ξ n ) ∈ R n−2 with respect to (n − 2)-dimensional Lebesgue measure, for the vector ξ = (ξ 1 , . . ., ξ n ) and some j 0 (ξ), the embedded sequence q j,1 e 1 + q j,2 e 2 + q j,3 e n+1 = (q j,1 , q j,2 , 0, . . ., 0, q j,3 is the tail of the sequence of best approximations. In fact, the logarithmic factor in ( 14) is only required when n = 3.Since the norms of best approximations grow exponentially [3], we see that the condition (14) holds as soon as for some ǫ > 0 we have It is not hard to see that this is in turn satisfied if ( 15) Note that any ξ ∈ R n arising from Theorem 4.1 is automatically totally irrational as soon as ξ 1 , ξ 2 has this property (otherwise the sequence of best approximations for ξ would terminate).
However, we will require the more general properties of Lemma 4.2 in place of (16).Conversely, the upper bounds (17) dim hold, where the non-obvious left estimates are again part of [18,Theorem 3].The full claim of [18,Theorem 3] summarizes the properties ( 16), ( 17), (18) in short as

Proof of Theorem 1.2
Let us immediately introduce a subset of Γ n of relevance below.
Definition 3. Let Γ n ⊆ R n be the set of ξ ∈ Γ n for which the three-dimensional lattice from the definition of R(ξ) can be chosen It is obvious that Γ 2 = Γ 2 which is just the set of totally irrational ξ ∈ R 2 , as well as

Proof of lower bounds.
A key observation is that the work of Das, Fishman, Simmons, Urbański [5, Here we implicitly restrict to (ξ 1 , ξ 2 ) totally irrational.They further provided a different, more complicated, explicit formula for w < 2 + √ 2 as well that we want to avoid stating.Formula (20), but not the formula for n = 3, follows alternatively from the independent paper [2] and Jarník's identity that relates one linear form with simultaneous approximation for two variables.
On the other hand, by the observations in § 4.1, via using Theorem 4.1, we may apply (I) of Lemma 4.2 with M = Γ n , and parameters n 1 = 2, t = n 2 = n − 2 and s = V(n).Note hereby that any arising ξ is indeed totally irrational by the concluding remark in § 4.1.By (19) and Lemma 4.2 we infer the lower bound √ 2 and after some simplifications of the according formula when n = 3 < 2 + √ 2, the right hand side becomes the respective values in (2).
Regarding packing dimension, it follows directly from [5, Theorem 3.10] that the packing dimension of the set involved in (20) is at least 1 for any w ≥ 2, with equality for w ≥ 3. Combined with (II) of Lemma 4.2 for the same parameters n i , t as above, indeed for n ≥ 3.For n = 2 the claim is obvious.Remark 3.An alternative proof of the bound for the packing dimension follows from (19) and the stronger claim dim P (Θ n ) ≥ n − 1 proved in § 9.5 below.

5.2.
Upper bounds: Special three-dimensional lattice.We first show the upper bounds for the smaller set Γ n where the three-dimensional lattice containing all large best approximations is just e 1 , e 2 , e n+1 Z .From the definition of Γ n and by Dirichlet's Theorem (4), we see that any Combined with (17), we get As previously noticed, by [5, Theorem 4.9] the right dimension is again V(n) as for the sets in (20) where strict inequality is imposed.This proves the reverse upper bound for the Hausdorff dimension of the sets Γ n .
Combining (21) with ( 18), we get that where the last identity is again due to [5, § 3.3].The reverse lower bound n − 1 for dim P ( Γ n ) (thus also for dim P (Γ n )) for n ≥ 2 was already shown in § 5.1, hence identity (3) is proved for the smaller sets Γ n .
5.3.Upper bounds: General case.We settle the upper bounds for the larger sets Γ n where the three-dimensional integer lattice is arbitrary.The main idea is to apply rational automorphisms of R n+1 to reduce it to the special case of § 5.2.Our proof below performing this in detail is reasonably lengthy and may not be the easiest available.
First notice that since there are only countably many three-dimensional sublattices of Z n+1 and by sigma-additivity of measures, it suffices to show that for any fixed threedimensional sublattice L of Z n+1 , the set of ξ ∈ R n inducing some tail of best approximations in L, has Hausdorff and packing dimension at most as in Theorem 1.2.Denote by Γ n (L) ⊆ Γ n ⊆ R n this set for any fixed given sublattice There is a bijective linear map f L : R n+1 → R n+1 induced by some integer matrix A L ∈ Z (n+1)×(n+1) that maps L to the particular sublattice e 1 , e 2 , e n+1 Z , as in Γ n .To see this, we extend a Z-basis of L to any vector basis of R n+1 consisting of integer vectors, then map the three Z-base vectors of L to e 1 , e 2 , e n+1 respectively, then extend it to an automorphism of R n+1 by mapping the remaining n − 2 integer base vectors to e 3 , . . ., e n , and finally multiply the arising rational matrix by the common denominator.Denote by g L : R n+1 → R n+1 the adjoint map of the inverse f −1 L of f L .Since g L is an automorphism as well, it is bi-Lipschitz and thus preserves Hausdorff and packing dimension [6].Hence if we write that just equals the ξ → ξ * map, we have We will bound the dimensions for the left hand side image sets.
Define an affine and a linear hyperplane of R n+1 , parallel to each other, by Define further a map ∆ : Note that ξ * = ι(ξ) ∈ A for any ξ ∈ R n and ∆ is just the identity on B. Obviously ∆ maps R n+1 \ B onto A. We claim that when restricting its domain to g L (A) \ B ⊇ g L (Γ L ) \ B, it is injective.Indeed, clearly g L (A) is an affine but not a linear subspace (as the image of an affine, non linear subspace under an automorphism).Thus it has only a singleton as intersection with any line through the origin, proving the claim in view of the definition of ∆.Hence, as Γ L ⊆ A and thus g L (Γ L ) ⊆ g L (A), and as ∆ is locally bi-Lipschitz on R n+1 \ B, writing g L (Γ L ) \ B as a countable union of sets with last coordinate bounded away from 0 in absolute value, by an easy sigma-additivity argument for measures, we have dim On the other hand, on B the map ∆ is just the identity.Thus together with (22) we easily conclude that Obviously the same identities hold when restricting the left hand side sets, containing only vectors with last coordinate either 0 or 1, to the first n coordinates (i.e.chopping off the last coordinate).In other words, if we let the projection that reverses ι, denoting this projected set by Therefore it suffices to bound from above the Hausdorff and packing dimensions of U L as in Theorem 1.2.
Let ξ ∈ Γ n (L) so that ξ * ∈ Γ L be arbitrary, and q ∈ L ⊆ Z n+1 .Then by definition of g L we have and by construction in particular it is an integer vector.Moreover, as bijective linear map f L is bi-Lipschitz so that (26) f L (q) ≍ q , with some absolute implied constants.Write g L (ξ * ) = (ζ 1 , . . ., ζ n+1 ) and let By (24) we have In any case, we get that where the implied constant is absolute on sets where Combining (26), ( 27) and as we may choose q best approximations for ξ and by definition of ζ, we get that ω(ζ) ≥ ω(ξ) ≥ n.
By the special form (25) of the integer vectors f L (q), it is further clear that the projection of ζ to the first two coordinates has the same property, i.e. ( For v > 1, define parametric subsets of U L given as where ζ n+1 is the last coordinate of g L (ξ * ) as above.Then ∆ is bi-Lipschitz on any X v .Thus again by the invariance of Hausdorff and packing dimension under bi-Lipschitz maps, for any v > 1, the quantities dim H (U L (v)) and dim P (U L (v)) can be estimated as in § 5.2 by precisely the same argument via (28) and ( 17), (18).Since we may write as a countable union of such sets, again by sigma-additivity of measures the same estimates hold for U L and finally in view of (23) for Γ n (L ) as well.
Remark 4. We cannot conclude that f L (q) ∈ Z n+1 are best approximations for ζ = π(∆(g L ((ξ * ))) ∈ R n .However, it suffices for the argument that they induce approximations of order at least n.

Sketch of the Proof of Theorem 1.3
By a slightly more general version of Theorem 4.1 from [14], again for any element of {(ξ 1 , . . ., ξ k−1 ) ∈ R k−1 : ω(ξ 1 , . . ., ξ k−1 ) > n}, we get some full measure set Here Γ n,k ⊆ Γ n,k is defined likewise as Γ n = Γ n,3 from § 5 with respect to the k-dimensional lattice L n,k := e 1 , . . ., e k−1 , e n+1 Z .Conversely, very similarly as in § 5.2 we get Combining these properties with (I) of Lemma 4.2 and (17) yield the claims ( 6), (7) on the Hausdorff dimension for the smaller sets Γ n,k .Very similarly as in § 5.3, via rational automorphisms that map a given k-dimensional rational lattice L ⊆ Z n+1 to L n,k , we lift the upper bound to the larger set Γ n,k .
Regarding packing dimension, we have as can be seen via [5, Theorems 3.8 & 4.9] with a short calculation.Thus, using the above observations on Γ n , we conclude with part (II) of Lemma 4.2 where n 1 = k − 1, n 2 = t = n − k + 1 (for lower bounds) and ( 18) (for upper bounds) that Finally again similarly as in § 5.3 we can lift the upper bound to the sets Γ n,k , the reverse inequality being a trivial consequence of (29), hence (5) holds.

Proof of Theorem 2.3
The lower bounds are clear by Theorem 2.1, we need to prove the upper estimates.Since there are only countably many finite subsets of sublattices of Z n+1 and by sigmaadditivity of measures, there is a subset Z n,k ⊆ Y n,k with the property that dim H (Y n,k ) = dim H (Z n,k ) and so that the finite collections of k-dimensional sublattices L 1 , . . ., L i(ξ) of Z n+1 are the same for any ξ ∈ Z n,k .So assume this set of lattices L 1 , . . ., L i is fixed.Clearly by pigeon hole principle for any ξ ∈ Z n there is some lattice L j , j = j(ξ) ∈ {1, 2, . . ., i} containing infinitely many best approximations.Again by additivity of measures, it suffices to treat the case where this lattice is the same for any ξ ∈ Z n,k .Without loss of generality we can assume it is L := L 1 .However, then within L we find infinitely many vectors inducing approximations of order > ω(ξ) − ε.If L = H 1 = e 1 , . . ., e k−1 , e n+1 Z , then it follows from Dirichlet's Theorem (4) that ω(ξ 1 , . . ., ξ k−1 ) ≥ ω(ξ) ≥ n for any ξ = (ξ 1 , . . ., ξ n ) ∈ Z n,k .Otherwise, we extend any rational linear map sending any base of L bijectively to e 1 , . . ., e k−1 , e n+1 to a rational automorphism of R n+1 and argue very similarly as in § 5.3 to get a set of the same Hausdorff dimension dim H (Z n,k ) where this is the case.Hence (17) and a formula generalising a classical result of Jarník [8] to higher dimension, see for example [1], imply

Proof of Theorem 2.1: Upper bounds
The upper bounds in ( 12) and ( 10) follow immediately from Theorem 1.2 and (8).We are left with the proof of (11).For this we use a different strategy.We show that for any If this is true then a classical metrical formula by Jarník [8] and (17) indeed imply Let q = (q 1 , q 2 , q 3 ) ∈ Z 3 be a best approximation of large norm for ξ.Without loss of generality we can assume q ∈ H 2 , thus q 1 = 0. Let µ be implictly defined by Then by Dirichlet's Theorem (4) and since q is a best approximation, we have µ ≥ 2. Then −q 3 /q 2 is a convergent to ξ 2 .Note further that by (32) and the theory of continued fractions, the next convergent has denominator at least q µ /2 (see [15,Proposition 5.2]).Hence, any integer linear form aξ 2 + b with max{|a|, |b|} < q µ /2 satisfies |aξ 2 + b| ≥ q −µ .In other words, there is no better approximation for ξ within H 2 up to norm q µ /2.On the other hand, by Dirichlet's Theorem there is some p = (p 1 , p 2 , p 3 By (32) clearly p = q.We may assume p is a best approximation for ξ, so since (ξ 1 , ξ 2 ) ∈ Θ 2 and the above argument excludes p ∈ H 2 , we must have p ∈ H 1 .Clearly p > q .Let r ∈ Z 3 be the best approximation for ξ following p.By a very similar argument as above based on Dirichlet's Theorem for p in place of q, we can now exclude r ∈ H 1 , so we must have r ∈ H 2 .But then r ≥ q µ /2 by the above observation.Hence there is no other best approximation between p and q µ /2, so p minimizes |u • ξ * | among all integer vectors u ∈ Z 3 with û < q µ /2.On the other hand, again by Dirichlet's Theorem has a solution v ∈ Z 3 .Again we can assume v is a best approximation, hence the above observation that there is no best approximation with norm in ( p , q µ /2) implies v = p.
Combining the right estimate from (34) with the left bound from (33) and this happens for infinitely many p ∈ H 1 as above, we see that ω(ξ 1 ) ≥ 4.An analogous argument yields ω(ξ 2 ) ≥ 4 as well, hence (30) is proved.
Remark 5.The argument in fact shows that the best approximations in H 1 and H 2 must occur at some high rate when (30) is close to optimal.Using the variational principle [4,5], with some effort some stronger bound in the interval (1, 7/5) can be obtained, however we omit its slightly techincal explicit calculation.

Remark 6. An analogous argument shows in general that
We can conclude and by ( 17) go on to estimate However, by [15, Theorem 3.3], for n ≥ 3 the right hand side in (35) is at least n − 1, in particular the right expression exceeds twice the single dimension of its factors.Hence it seems the bound in ( 12) cannot be reached with this method.This argument is most likely true for n = 2 as well (see [15,Conjecture 3]), so just (30) may be insufficient to improve on (31) either and the bound 1 seems to be the optimal outcome of the method.9. Proof of Theorem 2.1: Lower bounds 9.1.Outline.We first show in § 9.2 that for n = 2, there exist vectors (ξ 1 , ξ 2 ) ∈ R 2 with properties (i)-(v), and with a slight twist of (vi).The transition to general n as well as the weaker lower bound n − 2 for the Hausdorff dimension in § 9.3 will then be an easy consequence of Theorem 4.1 above obtained in [14].By modifications of the method, the stronger metrical claims will be proved in § 9.4, 9.5.

Existence claim:
Case n = 2.In this section, we prove.
Theorem 9.1.There exist uncountably many totally irrational ξ ∈ R 2 for which the tail of the best approximation sequence with respect to the maximum norm lies in the union of the two 2-dimensional sublattices of Z 3 given by Moreover, large best approximations alternately lie in H 1 and H 2 .Furthermore, for any given w ∈ [2, ∞] we can choose ξ so that additionally ω(ξ) = w.
Note that the condition ω(ξ) = w 2 in (vi) is missing for a full analogue of Theorem 2.1.Indeed, the vectors ξ constructed in this section satisfy ω(ξ) = w 2 − 1 instead.We construct our real vector.Let τ > 1 + √ 2 be a parameter.Let α j , β j be increasing positive integer sequences and derive the integers Clearly such choices are possible.Then in particular (37) Then, upon changing initial terms if necessary, we can assume We claim that it satisfies the assertions of the theorem. Put Then F j ≡ 1 mod 2 and G j ≡ 1 mod 3 imply the coprimality assertions Then obviously Thus by ( 38) clearly Then by (37) moreover , are small linear form for j ≥ 1, when τ is large.We show that v j and w j precisely comprise all large best approximations.This obviously finishes the proof.
Let b be any best approximation.Then by (41) there is an index j such that either vj ≤ b < ŵj or ŵj ≤ b < vj+1 .We show that in the first case b = ±v j , and in the latter case b = ±w j .Assume the first case, so the latter works very similarly by symmetry.First observe that since b is a best approximation of norm at least vj , we know that We distinguish two cases.
Case 1: b lies in the two-dimensonal subspace of R 3 spanned by v j , w j , i.e. b ∈ v j , w j R ∩ Z 3 .The special form of A j , B j and (39) imply the following crucial result on integer vectors in the two-dimensional lattices v j , w j R ∩ Z 3 .Proposition 9.2.For v j , w j as above, if a linear combination gv j + hw j is an integer vector, then in fact g ∈ Z and h ∈ Z.In other words, v j , w j R ∩ Z 3 = v j , w j Z .
Proof.Clearly we must have g, h ∈ Q.If we write (p 1 /q 1 )v j + (p 2 /q 2 )w j with p i /q i in lowest terms, then it is clear that q 1 must be a non-negative integer power of 2 and q 2 a non-negative integer power of 3 to make the first two coordinates (p 1 /q 1 )A j = (p 1 /q 1 )2 α j resp.(p 2 /q 2 )B j = (p 2 /q 2 )3 β j of gv j + hw j integers.But then by (39) clearly the third coordinate (p 1 /q 1 )F j + (p 2 /q 2 )G j is not an integer unless q 1 = q 2 = 1.
By the proposition applied to b and since b < ŵj obviously we must have h = 0. Hence b = gv j is an integer multiple of v j , but since b is a best approximation and thus primitive this integer must be g = ±1.Hence indeed b = ±v j .The case ŵj ≤ b < vj+1 works very similarly by symmetry and yields for the best approximation the only candidates ±w j .
Case 2: b does not lie in the space spanned by v j , w j .For this case we use an easy consequence of Minkowski's Second Convex Body Theorem.Lemma 9.3.There exists a constant c > 0 such that for any ξ ∈ R 2 and any parameter Q ≥ 1, the system Proof.Consider the integer lattice Z 3 and the box of (x 1 , x 2 , x 3 ) ∈ R 3 with coordinates It has volume 8c, independent of Q and ξ 1 , ξ 2 .Hence, by Minkowski's Second Convex Body Theorem, the product of the induced successive minima is ≪ c, hence choosing c small enough the third successive minimum is smaller than 1.This means there cannot be three linearly independent integer points within the box, which in turn is equivalent to the claim.
We first notice that both v j and w j induce approximations of order greater than two.By (40), (42) and as our choice of τ > 1 + √ 2 that implies (τ 2 − 1)/τ > 2, for some ǫ = ǫ(τ ) > 0 we get (46) and for w j by (43) we have a stronger estimate that also yields (47) Combined with (41), ( 44), (45), we have By the assumptions of Case 2, the three vectors v j , w j , b are linearly independent.So we get a contradiction to Lemma 9.3 for Q = ŵj , as soon as ŵj is sufficiently large.Hence in total Case 2 provides only finitely many best approximations, of small norm.
Combining our observations from Case 1 and Case 2, we see that v j and w j comprise all best approximations of large enough norm, as desired.Moreover it is clear that any ξ as above is totally irrational and by the freedom in the choice of A i , B i the set of induced ξ is uncountable.Finally it is easy to check from (46), (47), the fact that v j , w j comprise all the best approximations and (40) that ( 48) So choosing τ > 1 + √ 2 appropriately, we can realize any uniform exponent in (2, ∞).Finally, small modifications of the construction allow for obtaining the endpoints 2 and ∞ as well.9.3.General case and lower bound dim H (Θ n ) ≥ n − 2. As indicated before, the extension to the general case works with Theorem 4.1.In view of the sufficient condition ( 15) and (48), the lower bound dim H (Θ n ) ≥ n − 2 follows by taking any ξ 1 , ξ 2 constructed in § 9.2 upon increasing τ if necessary, and extending it to n-dimensional real vectors via Theorem 4.1 (we do not need Lemma 4.2 here).As noticed in § 4.1 any arising ξ ∈ R n is automatically totally irrational since (ξ 1 , ξ 2 ) has this property.In fact the same bound holds when restricting to vectors with arbitrary uniform exponent ω(ξ) = w ∈ [n, ∞], similar as in Θ n (w) but with some altered value for the ordinary exponent ω(ξ) in terms of w.
To improve the bound n − 2, in the next section we generalize the construction of § 9.2 to obtain some Cantor type set with the properties of Theorem 2.1, and determine a stronger lower bound for its Hausdorff dimension using results from [16,4,5].9.4.Proof of (9), up to strictness.In this section, we show the improved lower bound and thus as w can be arbitrarily close to n also dim H (Θ n ) ≥ n − 2 + 1/(n 2 + 1).Up to the latter inequality not being strict yet, this agrees with claim (9) in Theorem 2.1.
For τ > n, we now consider modified ξ 1 , ξ 2 , with sequences of continued fraction convergents (r i,j /s i,j ) j≥1 , i = 1, 2, with the following denominator growth properties: We then follow the proof above with A i resp.B i replaced by s 1,j resp.s 2,j , and F i resp.G i replaced by r 1,j resp.r 2,j , and we replace v j , w j by (52) In order to establish an analogue of Proposition 9.2, we need coprimality conditions for the denominators, concretely it suffices to guarantee Write K(τ ) for the set of (ξ 1 , ξ 2 ) ∈ R 2 satisfying (50), ( 51), (53).
Note that now by the theory of continued fractions 2,j , slightly stronger than in § 9.2 where the approximations were of order τ 2 − 1. Together with (50) it follows easily that ω(ξ 1 , ξ 2 ) ≥ τ 2 /τ = τ .Thus if τ > n, we can apply Theorem 4.1 again to see that for any (ξ 1 , ξ 2 ) ∈ K(τ ) there is a full measure set F n (ξ 1 , ξ 2 ) ⊆ R n−2 , so for any (ξ 3 , . . ., ξ n ) ∈ F n (ξ 1 , ξ 2 ) the vector (ξ 1 , . . ., ξ n ) has essentially the same best approximations as (ξ 1 , ξ 2 ) (with zeros added for entries at positions 3, 4, . . ., n).As in Case 1 in § 9.2, via an analogous claim to Proposition 9.2 for g j , h j , we can conclude that any best approximation within the space spanned by a pair g j , h j or a pair h j , g j+1 is actually equal to g j or h j .Moreover, similar to Case 2 in § 9.2 by ω(ξ 1 , ξ 2 ) ≥ τ > n ≥ 2 via Lemma 9.3 we see that large best approximations must lie in such spaces, hence the large best approxmations are precisely the g j , h j .
Thus we have that the g j and h j again comprise all best approximations for any ξ as above, i.e. (ξ 1 , ξ 2 ) ∈ K(τ ) and (ξ 3 , . . ., ξ n ) ∈ F n (ξ 1 , ξ 2 ).Moreover it is clear that hence ω(ξ) = τ and ω(ξ) = τ 2 , so (vi) holds as well.Thus any vector of the form To finish the proof of (49), we identify w with τ and show As τ can be taken arbitrarily close to n, the claimed lower bound for Θ n follows as well.
Then, since the projection of a set has at most the Hausdorff dimension of the original set (by Lipschitz property of projections [6]), the claim (54) follows via We verify (55) to finish the proof.We need to show that for any ξ 1 as above (i.e. with condition (51) for i = 1), there exists ξ 2 ∈ R as above, that may depend on ξ 1 , i.e. so that conditions (50) and (53) hold as well (then (51) holds for i = 2 as well; in fact there is set identity in (55)).So let arbitrary ξ 1 with property (51) for i = 1 with convergent sequence (r 1,j /s 1,j ) j≥1 be given.Assume we have constructed the partial quotients of ξ 2 up to convergent r 2,j−1 /s 2,j−1 = [a 2,0 ; a 2,1 , . . ., a 2,j−1 ] for given j, and let a 2,j be the next partial quotient for ξ 2 to be fixed.We may assume s 2,j−1 < s 1,j in view of (50).From the recursion (57) s 2,j−2 + a 2,j s 2,j−1 = s 2,j for convergent denominators, there are many consecutive partial quotients a 2,j that induce s 2,j−2 + a 2,j s 2,j−1 = s 2,j ≍ s τ 1,j , as we need for (50) in the current step.We must show some of them induce (53) as well.
On the other hand, by (51) for i = 1, there are ≪ log(s 1,j s 1,j+1 ) ≪ log s 1,j many primes dividing either s 1,j or s 1,j+1 .Denote S j = {p ∈ P : p|(s 1,j s 1,j+1 )} the set of such primes.For condition (53) to hold in the current step, it suffices that any p ∈ S j does not divide s 2,j .Now, for any such prime p ∈ S j , again by the recursion (57) and (s 2,j−2 , s 2,j−1 ) = 1 we see that at most one congruence class modulo p for a 2,j will induce p|s 2,j .Hence by Chinese Remainder Theorem and estimate (58), there exist many partial quotients a 2,j for which s 2,j is not divisible by any such prime.If this is at least 1 for large j, we are done.However, again by (51) and a standard estimate for the number of prime divisors of an integer, the cardinality of S j can be bounded Hence we may estimate the latter factor of (59) which gives p∈S j (1 − p −1 ) ≪ exp(− log log s 1,j ) = (log s 1,j ) −1 .
Hence as δ > 0 indeed there remain many suitable a 2,j ∈ S j in each step so that condition (53) holds as well, and thus (55) is true.We have proved the claims of the section.
Remark 7. The right equality of (56) and Lemma 4.2, and as it is reasonable to expect that condition (53) is metrically negligible, suggest the bound n − 2 + 2/(n 2 + 1) stated in Conjecture 1.A slightly stronger conjectural bound, still of order n − 2 + 2n −2 − O(n −4 ) for large n, is motivated in § 9.5 below.9.5.Proof of strict inequality in (9) and lower bounds in (10).A small improvement of the bound compared to § 9.4 (and likewise presumably for Conjecture 1) can be made when extending the sets K(τ ) ⊆ R 2 to larger sets.We then instead of the formula from [16] apply the variational principle from [4,5] for m = n = 1 (approximation to a single real number) to estimate their Hausdorff and packing dimension, and finally conclude with Theorem 4.1 and Lemma 4.2 again.We use the template formalism from [4,5].The general definition of templates can be found in [5,Defintion 4.1], however in our easiest setting, a template f = (f 1 , f 2 ) consists just two piecewise linear, continuous functions f i (t) : [0, ∞) → R with the following properties: for all t ≥ 0, slopes among {−1, 0, 1}, where 0 is only possible on intervals where f 1 (t) = f 2 (t) = 0.Moreover, any local maximum of f 1 is a local minimum of f 2 , hence f 1 (t) = f 2 (t) = 0 at such points.
We evaluate the lower limit δ(f τ,ǫ ) of the average local contraction rate in t ∈ [0, T ] as T → ∞ according to the variational principle [5,Theorem 4.7].As ǫ → 0, a short calculation and (61) lead for any τ > n to the bound As τ > n can be arbitrary, we get Note that inserting τ = n, we obtain the left bound n − 2 + 1/(n 2 + 1) of ( 9).The maximum over τ is taken at some slightly larger value, thereby confirming the strict inequality in (9).
For the packing dimension, as ǫ → 0 we evaluate the upper limit δ(f τ,ǫ ) of the average contraction rates for the template in Figure 1 as The quantity tends to 1 as τ → ∞.Thus, for any ε > 0 (note ε = ǫ), choosing τ large enough and ǫ > 0 small enough, by means of (II) from Lemma 4.2 and as coordinate projections as Lipschitz maps again do not increase the packing dimension of a set [6], the variational principle [5,Theorem 4.7] gives the lower bound As ε > 0 can be arbitrarily small, we obtain the desired bound n − 1.
Remark 8.As remarked in the proof, for ξ ∈ R n obtained via ξ 1 , ξ 2 as constructed above, we have Thus, it is possible to find lower bounds for the Hausdorff and packing dimension of sets with a slightly altered definition compared to Θ n (w), for example in place of (vi) just restricting to ω(ξ) = w and omitting the claim on the ordinary exponent.
similar argument a lattice of dimension k is insufficient as well: As we find infinitely many best approximations in each H i , thus any lattice containing a tail of best approximations must have 2-dimensional intersection with each of them, by Remark 1 again.But this is only possible if its dimension exceeds k.Moreover, the analogue of (48) that reads holds.Since k = n, we have found a suitable vector (ξ 1 , . . ., ξ n ) = (ξ 1 , . . ., ξ k ).Now assume n > k.Increase τ if necessary, so that in addition to (64), we have (τ k − 1)/τ > n as well.Take any (ξ 1 , . . ., ξ k ) as in the case n = k above.We see that Hence we may apply a more general version of Theorem 4.1 from [14], stating that for almost all vectors (ξ k+1 , ξ k+2 , . . ., ξ n ) ∈ R n−k with respect to Lebesgue measure, the magnified vector ξ = (ξ 1 , . . ., ξ n ) has the same tail of best approximations up to lifting, i.e. again lie in the according lattices H i = H i (n).The claims (ii * ), (vii) follow analogously to the case n = k.The lower bound n − k for the Hausdorff dimension is clear, finally the inequality being strict can be shown by the method from § 9.4, we omit details.
11. Proof of Theorem 3.1 We first point out that in § 9.2 above, essentially we only require the special form of ξ 1 , ξ 2 to exclude in Case 1 other best approximations within the lattices v j , w j Z and w j , v j+1 Z , sublattices of e 1 , e 2 , e n+1 Z .In context of Theorem 3.1, this will not be an issue when we simply lift the sequence of best approximations of some (ξ 1 , ξ 2 ) to Z n+1 .However, it turns out that our method below requires the property ω(ξ 1 , ξ 2 ) > 3n − 4 to eliminate the dependence of the derived set F n from ξ 1 , ξ 2 .So let ξ 1 , ξ 2 be any such numbers, totally irrational.To simplify the argument below, we fix any constant (65) σ ∈ (3n − 4, ω(ξ 1 , ξ 2 )).
Note that any v j lies in the fixed three-dimensional sublattice e 1 , e 2 , e n+1 Z of Z n+1 as in the definition of Γ n .
Here we cannot use Theorem 4.1, so we follow another strategy.Again assume b is any best approximation of large norm.There is a unique j such that (67) vj ≤ b < vj+1 .
We will show that b = v j to finish the proof.First observe that since b is a best approximation of norm at least vj , we know that Besides, a simple application of Minkowski's Second Convex Body Theeorem implies Lemma 11.1.Let ξ 3 , . . ., ξ n satisfy (66) and ε > 0. For any Q ≥ Q 0 (ε), there exist n − 1 linearly independent integer vector solutions u 1 , . . ., u n−1 in Z n−1 to where again ûi omits the last coordinate of u i .
Remark 9.If we would restrict to c-badly approximable vectors in R n−2 whose Hausdorff dimension tends to n − 2 as c → 0 + , then we could sharpen the right hand side estimate to ≪ c,n Q −(n−2) , which would simplify the proof below a little.
Proof.Consider the integer lattice in Z n−1 and for Q ≥ 1 the convex body consisting of (x 1 , . . ., x n−1 ) ∈ R n−1 with max 1≤i≤n−2 The volume is 2 n−1 , so by Minkowski's Second convex body theorem the product of the successive minima are of order ≍ n 1 as well.The condition (66) tells us that the first successive minimum is ≫ Q −ǫ for any ǫ > 0 and all Q ≥ Q 0 (ǫ).Hence all successive minima are ≪ n Q ǫ/(n−2) .This easily yields the claim by slightly modifying ǫ to some ε if necessary.
does not have three linearly independent solutions in integer vectors b = (b 1 , b 2 , b 3 ).
4.2.Metric results for Cartesian products and fibers.Part (I) of the following is partial claim of [12, Proposition 2.3], originally due to Marstrand [11] when n 1 = n 2 = 1, see Federer [7, § 2.10.25] for arbitrary dimension.Part (II) can be obtained by slightly generalizing the proof of part (c) in the proof of Tricot's [18, Theorem 3].
Lemma 4.2 (Marstrand; Tricot).Let n 1 , n 2 be positive integers and M ⊆ R n 1 +n 2 be measurable.Denote fibers by M