Remarks on sums of reciprocals of fractional parts

The Diophantine sums $\sum_{n=1}^N \| n \alpha \|^{-1}$ and $\sum_{n=1}^N n^{-1} \| n \alpha \|^{-1}$ appear in many different areas including the ergodic theory of circle rotations, lattice point counting and random walks, often in connection with Fourier analytic methods. Beresnevich, Haynes and Velani gave estimates for these and related sums in terms of the Diophantine approximation properties of $\alpha$ that are sharp up to a constant factor. In the present paper, we remove the constant factor gap between the upper and the lower estimates, and thus find the precise asymptotics for a wide class of irrationals. Our methods apply to sums with the fractional part instead of the distance from the nearest integer function, and to sums involving shifts $\| n \alpha + \beta \|$ as well. We also comment on a higher dimensional generalization of these sums.


Introduction
The subject of this paper is the asymptotic behavior of the Diophantine sums for various irrational α, where • denotes the distance from the nearest integer function.These sums appear in many different fields such as uniform distribution theory [21,28], multiplicative Diophantine approximation [4,13], lattice point counting in polygons [2,9,22,23,31], dynamical systems [10,15,16,17] and random walks [6,7,8,33].We refer to [3,4] for a comprehensive survey.The behavior of the more general sum N n=1 n −p nα −q is highly sensitive to the value of the exponents p, q ≥ 0 [27].Sharp estimates in the case p = q = 2 were given in [2], and in the case p = q > 1 in [11].The case p > q leads to convergent series for certain irrationals [12].For higher dimensional generalizations of the sums in (1), see [1,19,20,29].
Hardy and Littlewood [22,23,24], Haber and Osgood [21], Kruse [27] and more recently Beresnevich, Haynes and Velani [4] gave estimates for the sums in (1) in terms of the Diophantine approximation properties of α that are sharp up to a constant factor.The main goal of the present paper is to remove the constant factor gap between the upper and the lower estimates, thereby establishing the precise asymptotics.
Recall that an irrational α is called badly approximable if inf n∈N n nα > 0. The best known estimates for such an α are 1 [4] N log N ≤ Our first result improves these. 1 For the sake of readability, with a slight abuse of notation in all error terms log x := log max{e, x}.
Theorem 1.Let α be a badly approximable irrational.For any N ≥ 1, with implied constants depending only on α.
The only previously known precise asymptotic result (without a constant factor gap between the upper and the lower estimates) for the sums in (1) appeared in an obscure paper of Erdős from 1948 [18], in which he showed that This result seems to have escaped the attention of several later authors who subsequently proved weaker estimates in the metric setting.We learned about (2) from the recent historical survey [3], and it served as the starting point of our investigations.Our next result improves (2) by finding the precise order of the error term.
Theorem 2. Let c > 0 be an arbitrary constant, and let ϕ be a positive nondecreasing function on with implied constants depending only on c, ϕ and α.If ∞ k=1 1/ϕ(k) = ∞, then for a.e.α the sets have upper asymptotic density 1.
Recall that the upper asymptotic density of a set A ⊆ N is defined as lim sup In particular, for a.e.α we have with any ε > 0, but these fail with ε = 0. Note that the cutoff nα ≥ c/N in the first sum is necessary, as for a.e.α we have nα < (n log n log log n) −1 for infinitely many n ∈ N.
Our main results, Theorems 3 and 4 on the sums in (1) with a general irrational α are presented in Section 2. We discuss closely related sums involving fractional parts and shifts nα + β , and comment on a higher dimensional generalization in Section 3.

Main estimates
For the rest of the paper, α is an irrational number with continued fraction α = [a 0 ; a 1 , a 2 , . ..] and convergents p k /q k = [a 0 ; a 1 , a 2 , . . ., a k ].Set s K = K k=1 a k .We refer to [26] for a general introduction to continued fractions.

General irrationals
In this section, we prove our main estimates for the sums in (1) with a general irrational α.Theorem 3. Let c > 0 be an arbitrary constant.For any K ≥ 0 and q The implied constants depend only on c.
and the claims trivially follow.We may thus assume that N > 4c.
Summing the identity nα Let R = 1 8 (s K + N/q K ) + c, and note that c/N ≤ R/N ≤ 1/2.The latter inequality follows from the assumption c < N/4 and the general estimate s K ≤ q K , which can be easily seen e.g. by induction on K.We will estimate the integral in (3) Consider now the integral on [R/N, 1/2].Let D N (α) denote the discrepancy of the point set {nα}, where the supremum is over all intervals I ⊆ [0, 1], and λ is the Lebesgue measure.In particular, A classical discrepancy estimate [28, p. 126] Finally, consider the integral on [c/N, R/N ].We will show that For any 1 ≤ n < n ′ ≤ N we have 1 ≤ n ′ − n < q K+1 , hence by the best rational approximation property of continued fraction convergents, ).The pigeonhole principle thus shows that The previous formula implies (6) whenever N ≫ q K+1 .It remains to prove (6) when, say, N < q K+1 /4.
Theorem 4. For any K ≥ 0 and q K ≤ N < q K+1 , N n=1 with a universal implied constant.

Corollaries
Theorems 3 and 4 establish the asymptotics of the sums in (1) for a large class of irrational α.For instance, we immediately obtain As a further example, consider Euler's number e = [2; 1, 2, 1, . . ., 1, 2n, 1, . ..].The convergent denominators grow at the rate log q k = (k/3) log k + O(k), and s K = K 2 /9 + O(K).Theorems 3 and 4 thus give We will need certain basic facts from the metric theory of continued fractions in order to deduce the a.e.asymptotics from Theorems 3 and 4. Khinchin and Lévy showed that log q k ∼ π 2 12 log 2 k for a.e.α, whereas Borel and Bernstein proved that given a positive function ϕ, for a.e.α we have a k ≥ ϕ(k) for infinitely many k ∈ N if and only if ∞ k=1 1/ϕ(k) = ∞.A theorem of Diamond and Vaaler [14] on trimmed sums of partial quotients states that We refer to the monograph [25] for the proof of all these results and for more context.
By the Diamond-Vaaler theorem (9), Theorem 4 thus yields +x is also positive and nondecreasing, and satisfies ∞ k=1 1/ϕ * (k) = ∞.By the Borel-Bernstein theorem, for a.e.α we have a K+1 ≥ ϕ * (K) ≥ K for infinitely many K. Theorem 3 gives that for infinitely many K and any 4(ca In particular, the upper asymptotic density of the set is at least 1 − 4c 1/2 ε.Since ε > 0 was arbitrary, the upper asymptotic density is 1, as claimed. Similarly, for a.e.α we have a K+1 = max 1≤k≤K+1 a k ≥ 2ϕ(100K) + K log K log log K for infinitely many K. Theorem 4 gives that for infinitely many K and any q In particular, the set has upper asymptotic density 1, as claimed.

Badly approximable irrationals
In the special case of a badly approximable α Theorems 3 and 4 yield the value of the sums in (1) up to an error O(N log log N ) resp.O(log N log log N ).We now show how to modify the proof to remove the factor log log N from the error terms.
Proof of Theorem 1.We have nα ≥ c/N for all 1 ≤ n ≤ N with a suitably small constant 0 < c < 1/2 depending only on α, hence The contribution of the integral on [1/2, ∞) is negligible: To estimate the integral on [c/N, 1/2], we use the local discrepancy estimates [30,Theorem 2] max with universal implied constants, where {•} is the fractional part function.In the special case of a badly approximable α these estimates immediately show that for all 1 ≤ N < q K+1 , In the last step we used the fact that k j=1 q j ≤ 3q k , and that there are ≪ log(B/A) + 1 convergent denominators q k that fall in any given interval [A, B].Since −α is also badly approximable, we can similarly estimate the number of 1 We mention that (10) can also be easily deduced from an explicit formula for the local discrepancy due to T. Sós [32].Hence Theorem 5. Let c > 0 be an arbitrary constant.For any K ≥ 0 and q If in addition 4(ca K+1 ) 1/2 q K ≤ N and K + 1 is odd, then The implied constants depend only on c.Further, for any K ≥ 0 and q K ≤ N < q K+1 , N n=1 with a universal implied constant.
Proof.This is a straightforward modification of the proof of Theorems 3 and 4. The parity conditions follow from the fact that p 2k /q 2k < α < p 2k+1 /q 2k+1 for all k ≥ 0.
The same holds for the sums with "odd" replaced by "even".Theorem 2 also remains true with nα replaced either by {nα}/2 or by (1 − {nα})/2.The proof is identical to that of Theorem 2; note that the Borel-Bernstein theorem holds without any monotonicity assumption on ϕ, so the parity conditions do not cause any difficulty.
A straightforward modification of the proof of Theorem 1 similarly shows that for any badly approximable α, and the same holds with {nα} replaced by 1 − {nα}.

Shifted sums
Some of our methods apply to shifted Diophantine sums as well.Here we only focus on the case of a badly approximable α, for which we find the precise asymptotics.All previously known results on shifted Diophantine sums have a constant factor gap between the upper and the lower estimates [4,5].
Theorem 6.Let α be a badly approximable irrational, and let β ∈ R. For any N ≥ 1, with an implied constant depending only on α, where with implied constants depending only on α and β.

A higher dimensional generalization
There are several natural higher dimensional generalizations of the sums in (1), which have been studied using Fourier analysis [1,29], geometry of numbers [4] and lattices [19,20].In this setting, a vector α ∈ R d is called badly approximable if inf where n ∞ = max 1≤k≤d |n k |.Here we only comment on a result of Fregoli [19], who showed that for any badly approximable vector α ∈ R d , His main result in fact yields the precise asymptotics of the previous sum, in particular giving an alternative proof of Theorem 1 based on lattices.
Theorem 7. Let α ∈ R d be a badly approximable vector.For any N ≥ 1, with implied constants depending only on α. uniformly in 0 < t ≤ 1/2.Using this instead of formula (10), the rest of the proof is identical to that of Theorem 1. Writing the second claim follows from summation by parts.