On sequences with prescribed metric discrepancy behavior

An important result of H. Weyl states that for every sequence $\left(a_{n}\right)_{n\geq 1}$ of distinct positive integers the sequence of fractional parts of $\left(a_{n} \alpha \right)_{n \geq1}$ is uniformly distributed modulo one for almost all $\alpha$. However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy $D_{N}$ of $\left(\left\{a_{n} \alpha \right\}\right)_{n \geq 1}$ for almost all $\alpha$. By a result of R. C. Baker this discrepancy always satisfies $N D_{N} = \mathcal{O} \left(N^{\frac{1}{2}+\varepsilon}\right)$ for almost all $\alpha$ and all $\varepsilon>0$. In the present note for arbitrary $\gamma \in \left(0, \frac{1}{2}\right]$ we construct a sequence $\left(a_{n}\right)_{n \geq 1}$ such that for almost all $\alpha$ we have $ND_{N} = \mathcal{O} \left(N^{\gamma}\right)$ and $ND_{N} = \Omega \left(N^{\gamma-\varepsilon}\right)$ for all $\varepsilon>0$, thereby proving that any prescribed metric discrepancy behavior within the admissible range can actually be realized.


Introduction
H. Weyl [12] proved that for every sequence (a n ) n≥1 of distinct positive integers the sequence ({a n α}) n≥1 is uniformly distributed modulo one for almost all reals α. Here, and in the sequel, {·} denotes the fractional part function. The speed of convergence towards the uniform distribution is measured in terms of the discrepancy, which -for an arbitrary sequence (x n ) n≥1 of points in [ 0, 1 ) -is defined by where A N ([ a, b) ) := # {1 ≤ n ≤ N | x n ∈ [ a, b)} . For a given sequence (a n ) n≥1 it is usually a very hard and challenging problem to give sharp estimates for the discrepancy D N of ({a n α}) n≥1 valid for almost all α. For general background on uniform distribution theory and discrepancy theory see for example the monographs [6,9].
A famous result of R. C. Baker [3] states that for any sequence (a n ) n≥1 of distinct positive integers for the discrepancy D N of ({a n α}) n≥1 we have for almost all α and for all ε > 0.
The first author is supported by a Schrödinger scholarship of the Austrian Science Fund (FWF). The second author is supported by the Austrian Science Fund (FWF): Project F5507-N26, which is part of the Special Research Program "Quasi-Monte Carlo Methods: Theory and Applications".
Note that (1) is a general upper bound which holds for all sequences (a n ) n≥1 ; however, for some specific sequences the precise typical order of decay of the discrepancy of ({a n α}) n≥1 can differ significantly from the upper bound in (1). The fact that (1) is essentially optimal (apart from logarithmic factors) as a general result covering all possible sequences can for example be seen by considering so-called lacunary sequences (a n ) n≥1 , i.e., sequences for which a n+1 an ≥ 1 + δ for a fixed δ > 0 and all n large enough. In this case for D N we have for almost all α (see [10]), which shows that the exponent 1/2 of N on the right-hand side of (1) cannot be reduced for this type of sequence. For more information concerning possible improvements of the logarithmic factor in (1), see [5].
Quite recently in [2] it was shown that also for a large class of sequences with polynomial growth behavior Baker's result is essentially best possible. For example, the following result was shown there: Let f ∈ Z [x] be a polynomial of degree larger or equal to 2. Then for the discrepancy D N of ({f (n)α}) n≥1 for almost all α and for all ε > 0 we have On the other hand there is the classical example of the Kronecker sequence, i.e., a n = n, which shows that the actual metric discrepancy behavior of ({a n α}) n≥1 can differ vastly from the general upper bound in (1). Namely, for the discrepancy of the sequence ({nα}) n≥1 for almost all α and for all ε > 0 we have which follows from classical results of Khintchine in the metric theory of continued fractions (for even more precise results, see [11]). The estimate (2) of course also holds for a n = f (n) with f ∈ Z [x] of degree 1. In [2] further examples for (a n ) n≥1 were given, where (a n ) n≥1 has polynomial growth behavior of arbitrary degree, such that for the discrepancy of ({a n α}) n≥1 we have ND N = O (log N) 2+ε for almost all α and for all ε > 0; see there for more details.
These results may seduce to the hypothesis that for all choices of (a n ) n≥1 for the discrepancy of ({a n α}) n≥1 for almost all α we either have This hypothesis, however, is wrong as was shown in [1]: Let (a n ) n≥1 be the sequence of those positive integers with an even sum of digits in base 2, sorted in increasing order; that is (a n ) n≥1 = (3, 5, 6, 9, 10, . . . ). Then for the discrepancy of ({a n α}) n≥1 for almost all α we have Interestingly, the precise value of κ is unknown; see [8] for the background.
The aim of the present paper is to show that the example above is not a singular counterexample, but that indeed "everything" between (3) and (4) is possible. More precisely, we will show the following theorem.
Then there exists a strictly increasing sequence (a n ) n≥1 of positive integers such that for the discrepancy of the sequence ({a n α}) n≥1 for almost all α we have

Proof of the Theorem
For the proof we need an auxiliary result which easily follows from classical work of H. Behnke [4].

Lemma 1.
Let (e k ) k≥1 be a strictly increasing sequence of positive integers. Let ε > 0. Then for almost all α there is a constant K (α, ε) > 0 such that for all r ∈ N there exist M r ≤ e r such that for the discrepancy of the sequence ({n 2 α}) n≥1 we have Proof. For α ∈ R let a k (α) denote the k-th continued fraction coefficient in the continued fraction expansion of α. Then it is well-known that for almost all α we have a k (α) = O (k 1+ε ) for all ε > 0. Let ε > 0 be given and let α and c (α, ε) be such that for all k ≥ 1.
Since q l ≥ 2 l 2 in any case, we have l ≤ 2 log q l log 2 , and we obtain (7) q l+1 ≤ c 1 (α, ε) q l (log q l ) 1+ε , for an appropriate constant c 1 (α, ε). In [4] it was shown in Satz XVII that for every real α we have .
Indeed, if we follow the proof of this theorem we find that even the following was shown: For every α and for every best approximation denominator q l of α there exists an Y l < √ q l such that Y l n=1 e 2πin 2 α ≥ c abs √ q l . Here c abs is a positive absolute constant (not depending on α).
Let now r ∈ N be given and let l be such that q l ≤ e r < q l+1 , and let M r := Y l from above. Then by (6) and (7) we obtain, for an appropriate constant c 2 (α, ε), By the fact that (see Chapter 2, Corollary 5.1 of [9]) which is a special case of Koksma's inequality, the result follows. Now we are ready to prove the main theorem.
Proof of the Theorem. Let (m j ) j≥1 and (e j ) j≥1 be two strictly increasing sequences of positive integers, which will be determined later. We will consider the following strictly increasing sequence of positive integers, which will be our sequence (a n ) n≥1 : The sequence (a n ) n≥1 is constructed in such a way that it contains sections where it grows like (n) n≥1 as well as sections where it grows like (n 2 ) n≥1 . By this construction we exploit both the strong upper bounds for the discrepancy of ({nα}) n≥1 and the strong lower bounds for the discrepancy of ({n 2 α}) n≥1 , in an appropriately balanced way, in order to obtain the desired discrepancy behavior of the sequence ({a n α}) n≥1 . In our argument we will repeatedly make use of the fact that Let α be such that it satisfies (5) with ε = 1 2 . Then it is also well-known (see for example [9]) that for the discrepancy D N of the sequence ({nα}) n≥1 we have (9) ND N ≤ c 1 (α) (log N) 3 2 for all N ≥ 2.
By the above mentioned general result of Baker, that is by (1), we know that for almost all α for the discrepancy D N of the sequence ({n 2 α}) n≥1 we have for all ε > 0 and for all N ≥ 2, for an appropriate constant c 3 (α, ε). Actually an even slightly sharper estimate was given for the special case of the sequence ({n 2 α}) n≥1 by Fiedler, Jurkat and Körner in [7], who proved that (10) ND N ≤ c 4 (α, ε) N Assume that α satisfies (10) with ε = 1 8 . Then (11) ND N ≤ c 2 (α) N 1 2 (log N) 3 8 for all N ≥ 2. Now for such α and for arbitrary N we consider the discrepancy D N of the sequence ({a n α}) n≥1 .
where D x,y denotes the discrepancy of the point set ({a n α}) n=x+1,x+2,...,y . Hence by (8), (9) and by the trivial estimate D B l−1 ≤ 1 we have for all l large enough, provided that (condition (i)) m l is chosen such that (log m l ) 2 ≥ E l−1 .

Case 2.
Let F l < N ≤ E l for some l. Then by Case 1 and by (8) and (11) we have for l large enough that We choose (condition (ii)) (12) e l := F l 2γ log F l 2γ .
Note that conditions (i) and (ii) do not depend on α. Now assume that l is so large that 2 (log F l ) 2 < F l γ 2 . Then F γ l 2 ≤ 2 (log F l ) 2 + (e l log e l ) 1 2 ≤ 2F γ l and (note that γ ≤ 1 2 ) (13) F l < N ≤ E l = F l + e l ≤ 2F l .

Case 3.
Let E l < N < F l+1 for some l. Then by Case 2 and by (8) and (9) we have for N large enough.
It remains to show that for every ε > 0 we have ND N ≥ N γ−ε for infinitely many N. Let l be given and let M l ≤ e l with the properties given in Lemma 1. Let N := F l + M l . Then by Lemma 1, Case 1, (8), (12) and (13) for l large enough we have This proves the theorem.