1 Introduction

Weyl [12] proved that for every sequence \(\left( a_{n}\right) _{n \ge 1}\) of distinct positive integers the sequence \(\left( \left\{ a_{n} \alpha \right\} \right) _{n \ge 1}\) is uniformly distributed modulo one for almost all reals \(\alpha \). Here, and in the sequel, \(\{ \cdot \}\) 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 \(\left( x_{n}\right) _{n \ge 1}\) of points in \(\left[ \left. 0,1\right. \right) \)—is defined by

$$\begin{aligned} D_N = D_{N} (x_1, \dots , x_N) = \underset{0 \le a < b \le 1}{\sup } \left| \frac{\mathcal {A}_{N}\left( \left[ \left. a,b\right) \right. \right) }{N} - \left( b-a\right) \right| , \end{aligned}$$

where \(\mathcal {A}_{N}\left( \left[ \left. a,b\right) \right. \right) := \# \left\{ 1 \le n \le N \left| \right. x_{n} \in \left[ \left. a,b\right. \right) \right\} .\) For a given sequence \(\left( a_{n}\right) _{n \ge 1}\) it is usually a very hard and challenging problem to give sharp estimates for the discrepancy \(D_{N}\) of \(\left( \left\{ a_{n} \alpha \right\} \right) _{n \ge 1}\) valid for almost all \(\alpha \). For general background on uniform distribution theory and discrepancy theory see for example the monographs [6, 9].

A famous result of Baker [3] states that for any sequence \(\left( a_{n}\right) _{n \ge 1}\) of distinct positive integers for the discrepancy \(D_{N}\) of \(\left( \left\{ a_{n} \alpha \right\} \right) _{n \ge 1}\) we have

$$\begin{aligned} ND_{N}=\mathcal {O} (N^{\frac{1}{2}} \left( \log N\right) ^{\frac{3}{2}+\varepsilon }) \quad \text {as }N \rightarrow \infty \end{aligned}$$
(1)

for almost all \(\alpha \) and for all \(\varepsilon >0\).

Note that (1) is a general upper bound which holds for all sequences \((a_n)_{n \ge 1}\); however, for some specific sequences the precise typical order of decay of the discrepancy of \((\{a_n \alpha \})_{n \ge 1}\) 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 \(\left( a_{n}\right) _{n \ge 1}\), i.e., sequences for which \(\frac{a_{n+1}}{a_{n}} \ge 1+\delta \) for a fixed \(\delta > 0\) and all n large enough. In this case for \(D_{N}\) we have

$$\begin{aligned} \frac{1}{4 \sqrt{2}} \le \underset{N \rightarrow \infty }{\lim \sup } \frac{ND_{N}}{\sqrt{2 N \log \log N}} \le c_{\delta } \end{aligned}$$

for almost all \(\alpha \) (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 \(f \in \mathbb {Z} \left[ x\right] \) be a polynomial of degree larger or equal to 2. Then for the discrepancy \(D_{N}\) of \(\left( \left\{ f (n) \alpha \right\} \right) _{n \ge 1}\) for almost all \(\alpha \) and for all \(\varepsilon > 0\) we have

$$\begin{aligned} N D_{N} = \Omega ( N^{\frac{1}{2}-\varepsilon } ). \end{aligned}$$

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 \((\{a_n \alpha \})_{n \ge 1}\) can differ vastly from the general upper bound in (1). Namely, for the discrepancy of the sequence \(\left( \left\{ n \alpha \right\} \right) _{n \ge 1}\) for almost all \(\alpha \) and for all \(\varepsilon > 0\) we have

$$\begin{aligned} ND_{N} = \mathcal {O}(\log N \left( \log \log N\right) ^{1+\varepsilon }), \end{aligned}$$
(2)

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 \(f \in \mathbb {Z} \left[ x\right] \) of degree 1. In [2] further examples for \(\left( a_{n}\right) _{n \ge 1}\) were given, where \(\left( a_{n}\right) _{n \ge 1}\) has polynomial growth behavior of arbitrary degree, such that for the discrepancy of \(\left( \left\{ a_{n} \alpha \right\} \right) _{n \ge 1}\) we have

$$\begin{aligned} ND_{N} = \mathcal {O}((\log N)^{2+\varepsilon }) \end{aligned}$$

for almost all \(\alpha \) and for all \(\varepsilon >0\); see there for more details.

These results may seduce to the hypothesis that for all choices of \(\left( a_{n}\right) _{n \ge 1}\) for the discrepancy of \(\left( \left\{ a_{n} \alpha \right\} \right) _{n \ge 1}\) for almost all \(\alpha \) we either have

$$\begin{aligned} ND_{N} = \mathcal {O} (N^{\varepsilon }) \end{aligned}$$
(3)

or

$$\begin{aligned} ND_{N} = \Omega (N^{\frac{1}{2}-\varepsilon }). \end{aligned}$$
(4)

This hypothesis, however, is wrong as was shown in [1]: let \(\left( a_{n}\right) _{n \ge 1}\) be the sequence of those positive integers with an even sum of digits in base 2, sorted in increasing order; that is \((a_n)_{n \ge 1} = (3,5,6,9,10,\dots )\). Then for the discrepancy of \(\left( \left\{ a_{n} \alpha \right\} \right) _{n \ge 1}\)for almost all \(\alpha \) we have

$$\begin{aligned} ND_{N} = \mathcal {O}(N^{\kappa +\varepsilon }) \end{aligned}$$

and

$$\begin{aligned} ND_{N} = \Omega (N^{\kappa - \varepsilon }) \end{aligned}$$

for all \(\varepsilon >0\), where \(\kappa \) is a constant with \(\kappa \approx 0.404\). Interestingly, the precise value of \(\kappa \) is unknown; see [8] for the background.

The aim of the present paper is to show that the example above is not a singular counter-example, but that indeed “everything” between (3) and (4) is possible. More precisely, we will show the following theorem.

Theorem 1

Let \(0 < \gamma \le \frac{1}{2}\). Then there exists a strictly increasing sequence \(\left( a_{n}\right) _{n \ge 1}\) of positive integers such that for the discrepancy of the sequence \(\left( \left\{ a_{n} \alpha \right\} \right) _{n \ge 1}\) for almost all \(\alpha \) we have

$$\begin{aligned} ND_{N} = \mathcal {O}(N^{\gamma }) \end{aligned}$$

and

$$\begin{aligned} ND_{N} = \Omega (N^{\gamma -\varepsilon }) \end{aligned}$$

for all \(\varepsilon >0\).

2 Proof of the theorem

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

Lemma 1

Let \(\left( e_{k}\right) _{k \ge 1}\) be a strictly increasing sequence of positive integers. Let \(\varepsilon > 0\). Then for almost all \(\alpha \) there is a constant \(K\left( \alpha , \varepsilon \right) > 0\) such that for all \(r \in \mathbb {N}\) there exist \(M_{r} \le e_{r}\) such that for the discrepancy of the sequence \((\{n^{2} \alpha \})_{n \ge 1}\) we have

$$\begin{aligned} M_{r} D_{M_{r}} \ge K\left( \alpha , \varepsilon \right) \sqrt{\frac{e_{r}}{\left( \log e_{r}\right) ^{1+\varepsilon }}}. \end{aligned}$$

Proof

For \(\alpha \in \mathbb {R}\) let \(a_{k} \left( \alpha \right) \) denote the kth continued fraction coefficient in the continued fraction expansion of \(\alpha \). Then it is well-known that for almost all \(\alpha \) we have \(a_{k}(\alpha ) = \mathcal {O} \left( k^{1+\varepsilon }\right) \) for all \(\varepsilon >0\). Let \(\varepsilon >0\) be given and let \(\alpha \) and \(c\left( \alpha , \varepsilon \right) \) be such that

$$\begin{aligned} a_{k} (\alpha ) \le c\left( \alpha , \varepsilon \right) k^{1+\varepsilon } \end{aligned}$$
(5)

for all \(k \ge 1\).

Let \(q_{l}\) the lth best approximation denominator of \(\alpha \). Then

$$\begin{aligned} q_{l+1} \le (c\left( \alpha , \varepsilon \right) l^{1+\varepsilon }+1) q_{l}. \end{aligned}$$
(6)

Since \(q_{l} \ge 2^{\frac{l}{2}}\) in any case, we have \(l \le \frac{2\log q_{l}}{\log 2}\), and we obtain

$$\begin{aligned} q_{l+1} \le c_{1} \left( \alpha , \varepsilon \right) q_{l} \left( \log q_{l}\right) ^{1+\varepsilon }, \end{aligned}$$
(7)

for an appropriate constant \(c_{1} \left( \alpha , \varepsilon \right) \). In [4] it was shown in Satz XVII that for every real \(\alpha \) we have

$$\begin{aligned} \left| \sum ^{N}_{n=1} e^{2 \pi i n^{2} \alpha } \right| = \Omega (N^{\frac{1}{2}}). \end{aligned}$$

Indeed, if we follow the proof of this theorem we find that even the following was shown: for every \(\alpha \) and for every best approximation denominator \(q_{l}\) of \(\alpha \) there exists an \(Y_{l} < \sqrt{q_{l}}\) such that \(\big | \sum ^{Y_{l}}_{n=1} e^{2 \pi i n^{2}\alpha } \big | \ge c_{abs } \sqrt{q_{l}}\). Here \(c_{abs }\) is a positive absolute constant (not depending on \(\alpha \)).

Let now \(r \in \mathbb {N}\) be given and let l be such that \(q_{l} \le e_{r} < q_{l+1}\), and let \(M_{r} := Y_{l}\) from above. Then by (6) and (7) we obtain, for an appropriate constant \(c_{2}\left( \alpha , \varepsilon \right) \),

$$\begin{aligned} \left| \sum ^{M_{r}}_{n=1} e^{2 \pi i n^{2} \alpha } \right|\ge & {} c_{abs } \sqrt{q_{l}}\\\ge & {} c_{2} \left( \alpha , \varepsilon \right) \sqrt{\frac{q_{l+1}}{\left( \log q_{l}\right) ^{1+\varepsilon }}}\\\ge & {} c_{2}\left( \alpha , \varepsilon \right) \sqrt{\frac{e_{l}}{\left( \log e_{l}\right) ^{1+\varepsilon }}}. \end{aligned}$$

By the fact that (see Chapter 2, Corollary 5.1 of [9])

$$\begin{aligned} M_{r} D_{M_{r}} \ge \frac{1}{4} \left| \sum ^{M_{r}}_{n=1} e^{2 \pi i n^{2} \alpha } \right| , \end{aligned}$$

which is a special case of Koksma’s inequality, the result follows. \(\square \)

Now we are ready to prove the main theorem.

Proof of Theorem 1

Let \((m_{j})_{j \ge 1}\) and \((e_{j})_{j \ge 1}\) 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 \(\left( a_{n}\right) _{n \ge 1}\):

$$\begin{aligned}&1,2,3, \ldots , \underbrace{m_{1}}_{ =: A_{1}},\\&A_{1}+1^{2}, A_{1} +2^{2}, A_{1}+3^{2}, A_{1}+4^{2}, \ldots , \underbrace{A_{1} + {e_{1}}^{2}}_{ := B_{1}},\\&B_{1}+1, B_{1}+2, B_{1}+3, \ldots , \underbrace{B_{1}+m_{2}}_{ =: A_{2}},\\&A_{2} +1^{2}, A_{2}+2^{2}, A_{2}+3^{2}, A_{2}+4^{2}, \ldots , \underbrace{A_{2} + {e_{2}}^{2}}_{ =: B_{2}},\\&B_{2}+1, B_{2}+2, B_{2}+3, \ldots , \underbrace{B_{2}+m_{3}}_{ =: A_{3}},\\&A_{3} +1^{2}, A_{3}+2^{2}, A_{3}+3^{2}, A_{3}+4^{2}, \ldots , \underbrace{A_{3} + {e_{3}}^{2}}_{ =: B_{3}},\\&\vdots \\ \end{aligned}$$

Furthermore, let

$$\begin{aligned} F_{s}:= \sum ^{s}_{i=1} m_{i} + \sum ^{s-1}_{i=1} e_{i} \quad \text {and} \quad E_{s} := \sum ^{s}_{i=1} m_{i} + \sum ^{s}_{i=1} e_{i}. \end{aligned}$$

The sequence \((a_n)_{n \ge 1}\) is constructed in such a way that it contains sections where it grows like \((n)_{n \ge 1}\) as well as sections where it grows like \((n^2)_{n \ge 1}\). By this construction we exploit both the strong upper bounds for the discrepancy of \((\{n \alpha \})_{n \ge 1}\) and the strong lower bounds for the discrepancy of \((\{n^2 \alpha \})_{n \ge 1}\), in an appropriately balanced way, in order to obtain the desired discrepancy behavior of the sequence \((\{a_n \alpha \})_{n \ge 1}\). In our argument we will repeatedly make use of the fact that

$$\begin{aligned} D_N (x_1, \dots , x_N) = D_N(\{x_1 + \beta \}, \dots , \{x_N + \beta \}) \end{aligned}$$
(8)

for arbitrary \(x_1, \dots , x_N \in [0,1]\) and \(\beta \in \mathbb {R}\), which allows us to transfer the discrepancy bounds for \((\{n \alpha \})_{n \ge 1}\) and \((\{n^2 \alpha \})_{n \ge 1}\) directly to the shifted sequences \((\{(M+n) \alpha \})_{n \ge 1}\) and \((\{(M+n^2) \alpha \})_{n \ge 1}\) for some integer M.

Let \(\alpha \) be such that it satisfies (5) with \(\varepsilon =\frac{1}{2}\). Then it is also well-known (see for example [9]) that for the discrepancy \(D_{N}\) of the sequence \(\left( \left\{ n \alpha \right\} \right) _{n \ge 1}\) we have

$$\begin{aligned} ND_{N} \le \overline{c}_{1} \left( \alpha \right) \left( \log N\right) ^{\frac{3}{2}} \end{aligned}$$
(9)

for all \(N \ge 2\).

By the above mentioned general result of Baker, that is by (1), we know that for almost all \(\alpha \) for the discrepancy \(D_{N}\) of the sequence \((\{n^{2} \alpha \})_{n\ge 1}\) we have

$$\begin{aligned} ND_{N} \le c_{3} \left( \alpha , \varepsilon \right) N^{\frac{1}{2}} \left( \log N\right) ^{\frac{3}{2}+\varepsilon } \end{aligned}$$

for all \(\varepsilon > 0\) and for all \(N \ge 2\), for an appropriate constant \(c_{3} \left( \alpha , \varepsilon \right) \). Actually an even slightly sharper estimate was given for the special case of the sequence \((\{n^2 \alpha \})_{n \ge 1}\) by Fiedler et al. [7], who proved that

$$\begin{aligned} ND_{N} \le c_{4} \left( \alpha , \varepsilon \right) N^{\frac{1}{2}} \left( \log N\right) ^{\frac{1}{4}+\varepsilon } \end{aligned}$$
(10)

for almost all \(\alpha \) and for all \(\varepsilon > 0\) and all \(N \ge 2\).

Assume that \(\alpha \) satisfies (10) with \(\varepsilon = \frac{1}{8}\). Then

$$\begin{aligned} ND_{N} \le \overline{c}_{2} \left( \alpha \right) N^{\frac{1}{2}} \left( \log N\right) ^{\frac{3}{8}} \end{aligned}$$
(11)

for all \(N \ge 2\). Now for such \(\alpha \) and for arbitrary N we consider the discrepancy \(D_{N}\) of the sequence \(\left( \left\{ a_{n} \alpha \right\} \right) _{n \ge 1}\).

Case 1 Let \(N=F_{l}\) for some l. Then \(ND_{N} \le E_{l-1} D_{E_{l-1}}+\left( N-E_{l-1}\right) D_{E_{l-1},F_{l}},\) where \(D_{x,y}\) denotes the discrepancy of the point set \(\left( \left\{ a_{n} \alpha \right\} \right) _{n=x+1, x+2, \ldots , y}\). Hence by (8), (9) and by the trivial estimate \(D_{B_{l-1}} \le 1\) we have

$$\begin{aligned} ND_{N}\le & {} E_{l-1} + \overline{c}_{1} \left( \alpha \right) \left( \log m_{l}\right) ^{\frac{3}{2}}\\\le & {} 2 \left( \log m_{l}\right) ^{2}\\\le & {} 2 \left( \log N\right) ^{2} \end{aligned}$$

for all l large enough, provided that [condition (i)] \(m_{l}\) is chosen such that \(\left( \log m_{l}\right) ^{2} \ge ~E_{l-1}\).

Case 2 Let \(F_{l} < N \le E_{l}\) for some l. Then by Case 1 and by (8) and (11) we have for l large enough that

$$\begin{aligned} ND_{N}\le & {} F_{l} D_{F_{l}} + \left( N-F_{l}\right) D_{F_{l},N}\\\le & {} 2 \left( \log F_{l}\right) ^{2} + \overline{c}_{2} \left( \alpha \right) \left( N-F_{l}\right) ^{\frac{1}{2}} \left( \log \left( N-F_{l}\right) \right) ^{\frac{3}{8}}. \end{aligned}$$

Note that \(0<N-F_{l} < e_{l}\).

We choose [condition (ii)]

$$\begin{aligned} e_{l} := \left\lceil \frac{{F_{l}}^{2 \gamma }}{\log \left( {F_{l}}^{2 \gamma }\right) }\right\rceil . \end{aligned}$$
(12)

Note that conditions (i) and (ii) do not depend on \(\alpha \). Now assume that l is so large that \(2 \left( \log F_{l}\right) ^{2} < \frac{{F_{l}}^{\gamma }}{2}\). Then

$$\begin{aligned} \frac{{F_{l}^{\gamma }}}{2} \le 2 \left( \log F_{l}\right) ^{2} + \left( e_{l} \log e_{l}\right) ^\frac{1}{2} \le 2 {F_{l}^{\gamma }} \end{aligned}$$

and (note that \(\gamma \le \frac{1}{2}\))

$$\begin{aligned} F_{l} < N \le E_{l} = F_{l} + e_{l} \le 2 F_{l}. \end{aligned}$$
(13)

Hence

$$\begin{aligned} ND_{N}\le & {} \max \left( 1, \overline{c}_{2} \left( \alpha \right) \right) 2 {F_{l}^{\gamma }}\\\le & {} \max \left( 1, \overline{c}_{2} \left( \alpha \right) \right) 2 N^{\gamma }. \end{aligned}$$

Case 3 Let \(E_{l} < N < F_{l+1}\) for some l. Then by Case 2 and by (8) and (9) we have

$$\begin{aligned} ND_{N}\le & {} E_{l} D_{E_{l}} + \left( N-E_{l}\right) D_{E_{l}, N}\\\le & {} 2 \max \left( 1, \overline{c}_{2} \left( \alpha \right) \right) {E_{l}^{\gamma }} + \overline{c}_{1} \left( \alpha \right) \left( \log \left( N-E_{l}\right) \right) ^{2}\\\le & {} 3 \max \left( 1, \overline{c}_{2} \left( \alpha \right) \right) N^{\gamma } \end{aligned}$$

for N large enough.

It remains to show that for every \(\varepsilon > 0\) we have \(ND_{N} \ge N^{\gamma -\varepsilon }\) for infinitely many N. Let l be given and let \(M_{l} \le e_{l}\) 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

$$\begin{aligned} ND_{N}\ge & {} M_{l} D_{F_{l}, N} -F_{l} D_{F_{l}}\\\ge & {} K\left( \alpha , \varepsilon \right) \sqrt{\frac{e_{l}}{\left( \log e_{l}\right) ^{1+\varepsilon }}}-2 \left( \log m_{l}\right) ^{2}\\\ge & {} \frac{{F_{l}^{\gamma }}}{\left( \log F_{l}\right) ^{3}} \\\ge & {} N^{\gamma -\varepsilon }. \end{aligned}$$

This proves the theorem. \(\square \)