On the multiplicative independence between n and ⌊ αn ⌋

In this article we investigate diﬀerent forms of multiplicative independence between the sequences n and ⌊ nα ⌋ for irrational α . Our main theorem shows that for a large class of arithmetic functions a, b : N → C the sequences ( a ( n )) n ∈ N and ( b ( ⌊ αn ⌋ )) n ∈ N are asymptotically uncorrelated. This new theorem is then applied to prove a 2-dimensional version of the Erd˝os-Kac theorem, asserting that the sequences ( ω ( n )) n ∈ N and ( ω ( ⌊ αn ⌋ ) n ∈ N behave as independent normally distributed random variables with mean log log n and standard deviation √ log log n . Our main result also implies a variation on Chowla’s Conjecture asserting that the logarithmic average of ( λ ( n ) λ ( ⌊ αn ⌋ )) n ∈ N tends to 0.


Introduction
exists.Even though d is not a probability measure, since it is not countably additive, it can be used to measure the size of infinite sets of integers.For instance, if we define A p to be the set of positive integers divisible by p ∈ P, then A p has density d(A p ) = 1/p.It is believed that the sequences (n) n∈N and (⌊αn⌋) n∈N , for α ∈ R\Q, have independent multiplicative structure.The following classical result of G. L. Watson [Wat53] from 1953 supports this claim, showing that the proportion of natural numbers n ∈ N such that n and ⌊αn⌋ are coprime is 6/π 2 , which is the same probability of two arbitrary integers being relatively prime.
In 1917, Hardy and Ramanujan [HR17] showed the following theorem which describes the asymptotic behaviour of the function ω(n) which counts distinct prime factors of n.
In 1940 P. Erdös and M. Kac proved their well-known result about the Gaussian law associated to the function ω, stated here.
Theorem 1.3 (see [EK40]).For all a, b ∈ R where a < b, Later in 2007, William D. Banks and Igor E. Shparlinski showed that one can replace n in the Erdős-Kac theorem by any Beatty sequence (see Definition 2.3).
Theorem 1.4 (see [BS07,Theorem 4]).Let α ∈ R be a positive irrational real number, then Our first result unifies the theorem of Banks and Shparlinski with the original result of Erdős-Kac, following the heuristic that the sequences n and ⌊αn⌋, for α irrational, behave independently.For our second result, we took inspiration from a long-standing conjecture in number theory attributed to Chowla.

(Logarithmically averaged Chowla
where λ is the Liouville function (see Definition 2.2) and E n N , E log n N are standard and logarithmic averages (see Definition 2.5).
Chowla's conjecture is known to be true only for k = 1, which case is elementally equivalent to the prime number theorem.In addition, due to recent developments, we know that the logarithmically averaged Chowla's conjecture holds for k = 2 [Tao16] and for odd values of k [TT18].Inspired by this recent breakthrough, we established the following result.Both Theorem A and Theorem B are consequences of a more general theorem about a class of bounded sequences a : N → C which we call BMAI (Bounded Multiplicative Almost Invariant).Among other things, this class of sequences includes all multiplicative functions which don't take the value 0; for more details and further examples, see Definition 2.23.
Theorem C. Let a : N → C and b : N → S 1 be BMAI sequences such that: Then, for any irrational α ∈ R, To derive Theorem A from Theorem C, we take a ) for certain functions F, G : N → C, and employ a generalization of the Erdős-Kac theorem on short intervals to verify that the hypothesis of Theorem C are satisfied for this choice of a and b.The details of this derivation are given in Section 3 and Section 4.
Likewise, for the proof of Theorem B we take a(n) = b(n) = λ(n) in Theorem C and use Theorem 4.5 and Lemma 4.6.For the details of the proof of Theorem B, we refer the reader to Section 3 and Section 4 as well.
Finally, since there are many intermediate steps involved in the proof of Theorem C, we include here a diagram illustrating the path to our main results.

Notation
First, we introduce some classical functions in number theory.
Definition 2.1 (Prime omega functions).We define the prime omega functions Ω, ω : N → N as follows.For n ∈ N, we write n = p a 1 1 • • • p a k k where p 1 , . . ., p k are distinct primes numbers and a j ∈ N for j = 1, . . ., k.We define In other words, ω(n) is the number of distinct prime divisors of n.We also define which is the numbers of prime divisors with multiplicity.
Definition 2.2 (Liouville function).The classical Liouville function is defined as Throughout the paper we will use e(x) := e 2πix , x ∈ R, for the complex exponential function.For a real number α, we will denote ⌊α⌋ the greatest integer less than or equal to α, and {α} = α − ⌊α⌋ the fractional part.We now define the notion of a Beatty sequence.
Definition 2.3 (Beatty Sequence).For α ∈ R + the corresponding Beatty sequence is the sequence of integers defined by Finally, we introduce some classical definitions to measure how fast a sequence goes to 0 and how bounded the growth rate of a sequence is.

Averaging
In our proofs, we will sometimes use expectation notation to denote averages.In particular, we define here the Cesàro and the logarithmic averages.
Definition 2.5.For a function f : N → C and N, S ∈ N, we define its • average as • logarithmic average as • logarithmic expectation over the primes as where p only takes prime values less than or equal to S.
Remark 2.6.When the limit lim N →∞ E n N (f ) exists, it is called the Cesàro average of the sequence f : N → C. In this case, the limit lim N →∞ E log n N (f ) exists as well.We will start by stating and proving results which relates the Cesàro average and the logarithmic average.
Proof.For each N ∈ N, define By Lemma 2.7 we have that: applying lim sup N yields lim sup as desired.
The following corollary will be useful later on.
Corollary 2.9.Let S ⊆ N such that |S| = ∞ and let f : N × S → C such that for all t ∈ S, f (•, t) is a bounded function.We have that lim sup The rest of this subsection will contain some useful known results.
The following lemmas, scattered throughout the literature, will allow us to transition between Cesàro averages and logarithmic expectations over the primes.
Lemma 2.11.For S, N ∈ N with N > S, let L(S) = p S 1 p , we have ).We also have Second, we estimate n N p,q S,p,q|n 1 n : n N p,q S,p,q|n 1 n = p S q S n N,p|n,q|n Finally, we estimate Now using our estimations, we get and as L(S) = O(log log S), we get the desired result.
Remark 2.12.We note that the above lemma still holds when dropping the condition N > S by observing that when p > N, n N p 1 n = 0.
Lemma 2.13.For any bounded function f : N → C, we have that (2.5) with lim sup S→∞ lim sup N →∞ ε(S, N) = 0. We also have that ) By Cauchy-Schwarz, and as f is bounded we get for some constant C independent of N and S. Now dividing by L(S), we get where the second term on the left can be written as The only thing left to show is that the error term goes to zero as desired.Indeed, lim sup where we used that L(S) = O(log log S) to conclude.
The proof for (2.6) is similar.
Lemma 2.14.For any bounded f : N → C and for any k, N ∈ N, we have Proof.By using the definition of logarithmic average, we have We will start by showing that the first part of the sum tends to zero as N goes to infinity.Indeed, we have n is bounded, and as log k log N −log k tends to 0 as N tends to infinity, we get that (2.7) goes to zero as N → ∞.
As for the second part of the sum, as f is bounded, we have that Using Lemma 2.13, we can derive the following orthogonality criterion, which has appeared in [Kát86], [DD82] and [BSZ13].
We also have the following logarithmic version of the inequality: where lim sup S→∞ lim sup We can expand Using the prime number theorem, we also have giving (3.2).Now let's prove (2.9): Next is a powerful theorem from [Tan60], formalizing the independent behavior of polynomials with regard to the function Ω.
Theorem 2.16 (see [Tan60, Theorem 1]).Let {f i : i k} be a finite family of pairwise relatively prime non-constant polynomials, where f i is a product of r i 1 irreducible polynomials.Let (2.11) Remark 2.17.Theorem 2.16 is also true when Ω is replaced by ω, proven in the same paper.

Almost Periodicity
We now recall the notion of almost periodic function, which plays an important role in the sequel.Given g : N → C, we define its 1-norm as Definition 2.18.Let f : N → C be an arithmetic function.We say that f is almost periodic (sometimes also referred to as Besicovitch almost periodic in the literature) if for each ε > 0 there is some linear combination h over C of exponential functions e(αn), α ∈ R, such that f − h 1 < ε.
Lemma 2.19.Let f : N → C and g : R → C be two functions such that f = g on N and g is a Riemann integrable bounded periodic function with period α ∈ R\Q.Then, for all ε > 0, ∃M ∈ N and c k ∈ C c k e(kx/α), Remark 2.20.If α ∈ Q, the result still holds.In fact, in that case f is periodic.
Proof.Consider the function h : R → C defined as h(x) = g(xα).Note that h is a 1-periodic function and piece-wise continuous.As h is bounded, it can be seen as a function of L 2 ([0, 1]).Thus, there exists c k := ĥ Take M big enough such that h− M k=0 c k e(kx) 1 ε, and define (2.12) As (nα −1 ) n is equidistributed in T (as α −1 is irrational), and |h−P | is a piece-wise continuous function on the torus, by the Weyl equidistribution criteria we have that 1 Therefore, applying limsup in (2.12), we conclude that f − P 1 ε.
Lemma 2.21.If α ∈ R, then ½ ⌊αZ⌋ : N → C is an almost periodic function, which can be approximated with linear combination of exponential functions e(αn) with α ∈ R\Q.
Proof.First, we note that for m ∈ Z, We note that ½ ⌊αZ⌋ = ½ Rα on N. Treating ½ Rα as a function over R, we have that it is a Riemann integrable bounded periodic function with period α ∈ R. Therefore, Lemma 2.19 yields the conclusion.
There is a Riemann integrable α-periodic function f over the reals such that f = ½ D p,i when restricted to N.
Proof.First, for any m ∈ N, (2.14) Therefore, we define the set and note that ½ D p,i = ½ D ′ p,i on N. Furthermore, f := ½ D ′ p,i is a Riemann integrable function, which is also α-periodic.

BMAI sequences
The goal of this subsection is to introduce the notion of BMAI sequences and list some properties.We let P denote the set of prime numbers. (2.15) The idea behind this definition is that the limiting behaviour of a BMAI sequence approximates that of a multiplicative function.
Remark 2.24.If a : N → C is a BMAI sequence we also have by Corollary 2.9 using S = P and for Example 2.25.The following are BMAI functions.
, for a bounded continuous F : R → C with θ p = 1, ∀p ∈ P.
• Any bounded multiplicative arithmetic function f , with θ p = f (p) nonzero, ∀p ∈ P. We prove some basic properties.
Proposition 2.26.Let a, b : N → C BMAI sequences.Then for primes p, q ∈ P we have and also lim sup Proof.We have the following inequality Corollary 2.27.BMAI sequences are closed under coordinate-wise multiplication, i.e. if (a(n)) n and (b(n)) n are BMAI then so is (a(n)b(n)) n .
Proof.Taking p = q, Proposition 2.26 gives lim sup Corollary 2.28.For a BMAI sequence a : N → C, for all p, q ∈ P.
Proof.Take b = a in Proposition 2.26.
Proposition 2.29.Let a : N → C be a BMAI sequence and f : N → C a bounded function.
Then for any and lim sup Proof.Without loss of generality, we can take f ≡ 1 since Note that Taking lim sup, we derive lim sup Proposition 2.30.Let a : N → C be a BMAI sequence, then there exists a (actual) multiplicative function θ : N → C such that for every m ∈ N lim sup Additionally, ||a|| 1 = 0 if and only if θ is unique.
Proof.Let (θ p ) p∈P be a sequence given by a being BMAI.We define θ : N → C, as the multiplicative function defined by θ(p) = θ p , ∀p ∈ P.
We prove the statement using induction over Ω(m).The definition of a BMAI sequence yields the case Ω(m) = 1.We assume that the statement holds for all m ∈ N with Ω(m) k.Let m ∈ N with Ω(m) = k and p ∈ P. We use Proposition 2.29 with d 1 = m, d 2 = 0 and f ≡ 1 to obtain lim sup Using (2.16) and the inductive hypothesis: and as every n ∈ N with Ω(n) = k + 1 can be written as n = pm for p ∈ P with p | n and m = n/p, thus concluding existence.
For uniqueness, take p ∈ P and let θ ′ p such that lim sup p , ∀p ∈ P. For the other direction, if ||a|| 1 = 0, then θ p ∈ C can be any value.In fact, notice that lim sup which doesn't depend on θ p .Lemma 2.31.Let a : N → C be a BMAI sequence such that for all natural numbers h ∈ N and irrational β, Then, ∀p, q ∈ P, β ∈ R\Q and h 1 , h 2 ∈ N with qh 1 − ph 2 = 0, lim sup E n N a(pn + h 1 )a(qn + h 2 )e(βn) = 0.

Basic Case
The following result is well known, as discussed in [JLW20], for multiplicative sequences.We generalize it to all BMAI sequences.
By lemma 2.28, we have that the first term of the sum is o p→∞ (1) + o q→∞ (1).And as β is irrational, we have that β(p − q) is not an integer and so lim M →∞ |E m M e(β(p − q)m)| = 0. Therefore, we have And so to prove that lim N →∞ By Lemma 2.21, ½ ⌊αN⌋ (m) is an almost periodic function such that for ε > 0, there is In addition, as d([αN]) = α −1 and for each β ∈ (0, 1), E n N e(mβ i ) → 0, we have that c 0 = α −1 .
Hence, we have lim sup By the assumptions on a, lim M →∞ 1 M M m=1 a(m)α −1 = α −1 c.By Lemma 3.1, the last term of the sum tends to 0, giving lim sup We conclude by taking ε ց 0.
Remark 3.3.Note that the condition of the above proposition can be weakened.Indeed, we can remove the conditions of a being BMAI and only ask for a bounded a : N → C which is orthogonal to e(nβ) for β irrational, i.e. lim M →∞ 1 M M m=1 a(m)e(mβ) = 0. Remark 3.4.By von Mangoldt [vM97, p. 852] and Landau [Lan09, pp.571-572, 620-621], the prime number theorem is equivalent to lim Corollary 3.5.For any irrational α > 1, we have and the fact that λ is BMAI, by Proposition 3.2, we obtain the desired result.

Main Theorem
We now state the main technical result, whose proof will come later.The main theorem follows from this result using Lemma 2.31.Theorem 3.6.Let a : N → C be a bounded sequence and b : N → S 1 be a BMAI sequence such that: 1. lim 2. for all β ∈ R\Q, p = q primes and (i,

Then for any irrational
We now use Theorem 3.6 to prove Theorem C.
The following example illustrates the necessity of the first condition in both theorems.
Example 3.7.Consider a ≡ b ≡ 1. Clearly both functions are BMAI with values in S 1 , and satisfy condition 2 of Theorem C by However, a clearly doesn't satisfy condition 1 as the associated limit is 1.
In contrast, the BMAI condition can be relaxed.Corollary 3.8 below is an example of a function which is not BMAI but satisfies the conditions of Theorem C. As for (3.5), note that one can approximate almost periodic functions by a finite sum of e(cn) functions.
The following lemma will be useful in proving Theorem 3.6.Lemma 3.10.For any bounded a : N → C that satisfies condition 1 of Theorem C, Proof.Consider the function and let 0 < ε < 1.By assumption, we can find an H ∈ N such that for every H H, lim sup In the following, we will omit the subscript H in γ H and just write γ.All statements will hold for any H H. Let E = {n ∈ N | γ(n) ε}, and note that Taking lim sup in Eq. (3.6), be the set of numbers n for which the H points after starting points Hn are covered by E up to ε.Note that Taking lim sup, we conclude that In other words, since E has high density, "good" starting points are also "dense".If n ∈ D then Hn is at most at εH distance of an element of m ∈ E.
for 0 k εH we conclude that γ(Hn) ε + 2ε a ∞ for every n ∈ D. Finally, we note that As this is valid for every H H, we conclude that lim H→∞ lim sup For the logarithmic version, we use Eq.(3.7) combined with Corollary 2.9, taking S = N and f (n, T ) = γ T (n).
Proof of Theorem 3.6.For simplicity, we restrict our attention to Cesaró averages, as the proof for logarithmic averages is analogous and all the necessary lemmas used in the proof hold for both types of averages.We will apply Lemma 2.15 to with lim sup S→∞ lim sup N →∞ ε(S, N) = 0. Similarly as in the proof of Proposition 3.2, we take limits and use Corollary 2.28 to get lim sup (3.8)Moreover, as |θ p θ q | C θ for all p = q, we get that lim sup (3.10) To begin, note that which tends to 0 as S goes to infinity.Thus, it remains to investigate the case where p, q > 1 α .Let i(p, n) := ⌊pαn⌋ − p⌊αn⌋, thus a(⌊pαn⌋)a(⌊qαn⌋) = a(p⌊αn⌋ + i(p, n))a(q⌊αn⌋ + i(q, n)).

Erdős-Kac generalization
For the following sections, we denote for every n ∈ N, and for a function F : R → C, First, we prove that a = Ψ G satisfies the conditions of theorem Theorem 3.6 whenever I(G) = 0. Proof.First, we show that Ψ G satisfies the first condition.Proving the following will be sufficient: Indeed, by expanding the squares and switching the order of summation, we get lim Splitting the sum into the diagonal terms and the non diagonal terms, The first sum tends to 0. Indeed let ε > 0 and (E i ) I i=1 be Jordan measurable sets, where E i ⊆ R for each i ∈ I, and (c i ) I i=1 ⊆ C I such that We may assume Notice that lim sup where we used Theorem 2.16.Hence lim sup Using Theorem 2.16 again, we have lim Using this, we derive lim sup Thus as ε > 0 was arbitrary, we get that lim For the second sum, notice that 1 for some absolute constant as G is bounded, and thus lim Hence the first condition is satisfied.
To verify the second condition, in light of Lemma 2.10, it is enough to show that for all h ∈ N lim (0, 0) and h > 0, we have that pn + ph + i, qn + qh + j, pn + i and qn + j are polynomials over Z that are pairwise relatively prime.We can thus apply Theorem 2.16 to get the desired result.
Then, for all a, b, c, d ∈ R, Proof.First, notice that the family of functions is such that • A separates points, in other words, for any x = y ∈ X, there exists f ∈ A such that f (x) = f (y), • for every x ∈ X, there exists f ∈ A such that f (x) = 0, Then by the Stone-Weierstrass theorem, A is dense in Thus, using the hypothesis, lim sup and as this is valid for all ε > 0, we conclude that lim and such that H is monotone in (a − ε, a + ε) and in (b − ε, b + ε).Therefore, we have that Taking lim sup and using Theorem 1.4 yields lim sup Therefore, as this is valid for every ε > 0, we conclude that lim and thanks to Erdős-Kac (Theorem 1.3) and Theorem 1.4, this is equivalent to lim concluding the proof.
Lemma 4.4.There exists a set E ⊆ N of density 1 such that Proof.Notice that log log⌊αn⌋ = log log n + b n , where (b n ) n is a bounded sequence (in which we used that log : (1, ∞) → R is 1-Lipschitz given that its derivative is bounded by 1), and let C n := w(⌊αn⌋)−log log n (log log⌊αn⌋) 1/2 .Note that Therefore, it is enough to see that |A n − C n | → n 0 with n taking values in a set of density 1.Using similar arguments, we have that (log log⌊αn⌋) We now restate and prove Theorem A. In order to prove Eq. (4.2), we show that lim Indeed, since G is bounded,

Logarithmic averages of the Liouville function
We now cite a relatively recent theorem of Matomaki and Radziwill.
Theorem 4.5 (see [MR16]).lim H→∞ lim sup Note that this is the first condition of our main theorem, Theorem C, satisfied by the Liouville function.The following lemma from [FH18] shows that the Liouville function and the Möbius function also satisfy the second condition of Theorem C. Proof.Immediate by taking b = λ and a = λ in Theorem C and using Lemma 4.6.

Open Questions
While we proved the relative independence of n and ⌊αn⌋ under specific conditions, we have yet to show independence of different Beatty sequences.Namely, one may consider ⌊αn⌋ and [βn⌋ instead, which leads to the following conjecture.Once we have pairwise independence of Beatty sequences, we can then consider a higherorder version.For example, the following conjecture offers a higher-order generalization of Theorem B.

Definition 2. 4 .
For a function g : N → R, which is strictly positive for all large enough values of n, we use the classic notation f (N) = o N →∞ (g(N)) to denote a function f : N → R denote f (N) = O g(N) , if there exists a positive real number C and an integer M ∈ N such that |f (N)| Cg(N), ∀N M.

Corollary 3. 8 .
For an irrational α and a real number c, Furthermore, for any almost periodic function f : N → C, )f (⌊αn⌋) = 0. (3.5) Remark 3.9.Similar results have been obtained in other contexts such as [GT12].Proof.If c ∈ Z, then (3.4) reduces to Conjecture 1.5 (Chowla's conjecture) for the case k = 1, which is well known to be equivalent to the prime number theorem.Suppose c ∈ R\Z, ∈ Z, lim H→∞ | 1 H H h=1 e(ch)| = 0 and e(cn) satisfies condition 1 and 2 of Theorem C. Since λ is a BMAI sequence, the result follows from Theorem C.
Lemma 4.1.Let G : R → C be a bounded continuous function such that I(G) = 0. Then Ψ G satisfies the conditions of Theorem C.

Theorem A .
Let α ∈ R be a positive irrational real number and a, b, c, d ∈ R. Thenlim N →∞ 1 N #{n N : a ω(⌊αn⌋) − log log n (log log n) 1/2 b, c ω(n) − log log n (log log n) 1/2 d} = 1 √ 2π b a e −t 2 /2 dt 1 √ 2π d c e −t 2 /2 dt .Proof.We want to show that limN →∞ E n N ½ [a,b] (A n )½ [c,d] (B n ) = I(½ [a,b] )I(½ [c,d] ).Using Lemma 4.3, it is enough to prove that for every continuous and compactly supported F : R → S 1 and bounded G : R → C with I(G) = 0, limN →∞ E n N F (A n )G(B n ) = 0. (4.2) .3) and as F is uniformly continuous and has compact support, it is sufficient to show|A n − B ⌊αn⌋ | − −− → n→∞ 0,with n taking values in a set of density 1, which is given by Lemma 4.4.In this way, it's enough to prove that for every continuous with compact support F : R → S 1 and bounded G : R → C with I(G) = 0 we have that limN →∞ E n N Ψ F (n)Ψ G (⌊αn⌋) = 0, (4.4)which follows from Lemma 4.2, concluding the proof.

Conjecture 5. 1 .
Let a : N → C and b : N → S 1 be a BMAI sequences such that we also have that1.lim H→∞ lim N →∞ n + h)| = 0 2. if for all β ∈ R\Q, h ∈ N we have that : lim N →∞ E n N a(n)a(n + h)e(nβ) = 0 ( resp E log n N ) Then for any irrational α, β ∈ R such that α and β are rationally independent, lim N →∞ E n N b(⌊βn⌋)a(⌊αn⌋) = 0. (resp E log n N )