On the distribution of the van der Corput sequences

For an integer p≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge 2$$\end{document}, let {xn}n∈N⊂T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{x_n\}_{n\in {\mathbb {N}}}\subset {\mathbb {T}}$$\end{document} be the p-adic van der Corput sequence. For intervals [0,α)⊂T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,\alpha )\subset {\mathbb {T}}$$\end{document} and for positive integers N, consider the geometrically-shifted discrepancy function Dp,N,α(t)=∑n=0N-1X[0,α)(xn+t)-Nα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{p,N,\alpha }(t)=\sum _{n=0}^{N-1}\mathcal {X}_{[0,\alpha )}(x_n+t)-N\alpha $$\end{document}. In this paper, we give a characterization of the asymptotic behavior of ‖Dp,N,α(·)‖L2(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}$$\end{document} for N→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\rightarrow \infty $$\end{document} that depends on the p-adic expansion of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}.


Introduction and statement of the result.
Let N denote the set of all nonnegative integers, and let T = R/Z denote the one dimensional torus. For every x ∈ R, let {x} represent the fractional part of x and let x represent the integer part of x. Also, set x = min({x}, 1 − {x}) as the distance from x to Z.
A sequence {x n } n∈N is said to be uniformly distributed in T if for any interval I ⊆ T with measure |I|, it holds We introduce the concept of discrepancy as a quantitative counterpart of the uniform distribution. Namely, for N ∈ N, we define the discrepancy of a sequence {x n } n∈N ⊂ T with respect to intervals as where X I : T → {0, 1} is the indicator function of the interval I ⊂ T.
In 1935, J.G. van der Corput conjectured that for any sequence of real numbers {x n } n∈N ⊂ T, the quantity D N ({x n } n∈N ) is unbounded with respect to N . This conjecture was first proved ten years later by van Aardenne-Ehrenfest [14] with a first lower bound, and in 1954, Roth [10] significantly improved this estimate with one of the most relevant results in discrepancy theory. Later in 1972, Schmidt [12] gave the best possible estimate (up to a multiplicative constant). Namely, he showed that for any sequence {x n } n∈N ⊂ T, it holds the lower bound lim sup We formally introduce the object of our studies. Let N + denote the set of all positive natural numbers, and consider p ≥ 2. For n ∈ N with p-adic expansion n = η j=1 n j p j−1 with η ∈ N + and 0 ≤ n j < p for all j, we define the p-adic van der Corput sequence {x n } n∈N ⊂ T as the sequence described by In 1935, van der Corput [15] showed that the discrepancy of the dyadic van der Corput sequence is optimal with respect to the order of magnitude in N , and it is well-known that the same holds for all p-adic van der Corput sequences. In particular, Faure [4] generalized p-adic van der Corput sequences and further investigated their qualities. Since its low-discrepancy property, the distribution of the van der Corput sequence has been studied under various points of view. We refer the reader to [6] for a detailed survey on all the generalizations of such sequence and the concerning properties.
We introduce a major tool in discrepancy theory. Consider a fixed anchored interval [0, α) ⊂ T and, for N ∈ N + , define the discrepancy function of the p-adic van der Corput sequence {x n } n∈N with respect to [0, α) as In 1972, Schmidt [11] first showed that if α has finite dyadic expansion, then D 2,N,α is bounded, and in 1978, Shapiro [13] proved the converse implication through ergodic theory methods. Later in 1980, Hellekalek [7] generalized the results for p-adic van der Corput sequences.
In 1983, Faure [5] gave an explicit formula for D p,N,α . For completeness, we report his result as follows. Let N ∈ N + , and write the p-adic expansions Further useful estimates for D 2,N,α can be found in [3]. It is now time to introduce the main tool of this paper.
For N ∈ N + , we define the geometrically-shifted discrepancy function of the p-adic van der Corput sequence {x n } n∈N with respect to [0, α) as One can interpret (1.1) as the usual discrepancy function of the p-adic van der Corput sequence with respect to a shifted interval [−t, −t + α) ⊂ T. In particular, D p,N,α (0) = D p,N,α .
The aim of this paper is to investigate the asymptotic behavior of D p,N,α (·) L 2 (T) , i.e., the root mean square average of the discrepancy function over all possible intervals of length α in T.
We remark that the quantity ⎛ is exactly the periodic L 2 -discrepancy introduced by Lev [9] in 1995. Moreover, we directly deduce from Lemma 2.2 that the latter quantity coincides with the diaphony introduced by Zinterhof [16] in 1976 (see [1,Proposition 2] for a generalization of such equality in higher dimensions). As a matter of fact, the periodic L 2 -discrepancy is a geometric interpretation of diaphony, and it is also closely related to the worst-case integration error of quasi-Monte Carlo integration rules (see, for example, [8]). We need a last definition in order to state our main result.
In other words, L p,N,α represents the set of the indices ∈ [1, η − 1] such that the coefficient α does not lie on the extremities {0, p − 1} or it belongs to {0, p − 1} but it is different from the previous coefficient. We further refer to Faure's [5, Theorem 1] on generalized van der Corput sequences for similarities between the quantities involved.
Then, in relation to the p-adic expansion of α, we can give an asymptotic characterization of D p,N,α (·) L 2 (T) with respect to N . We state the main result of this paper.

Theorem 1.3.
For α ∈ (0, 1) and p ∈ N + , we have It directly follows a characterization of the intervals for which the L 2 -norm of the geometrically-shifted discrepancy function is bounded.
2. Auxiliary results for the proof. From now on, let {x n } n∈N denote the p-adic van der Corput sequence. Moreover, improper sums (as 0 i=1 1) will be conventionally considered as zeros.
Notice that for any η ∈ N, we have More precisely, for any η ∈ N, the first p η points in the p-adic van der Corput sequence are p η equidistant points in T and, trivially, the point 0 is the first of them. We extend this fact in the following lemma.
Proof. We can uniquely write all the numbers k such that This consequently means that all the elements can uniquely be written as The first sum is then a constant term for all these x k , while the second sum is the same one that appears for the first p h−1 elements of the van der Corput sequence. Therefore Now, we need to recall some fundamental properties of Fourier analysis. For f ∈ L 2 (T), we define the Fourier transform of f as the function Ff : Z → C described by For y ∈ R, we use the notations τ y f = f (· − y) and e y (k) = exp(2πiyk). It is then a well-known property of Fourier transforms that F(τ y f ) = e −y Ff . Therefore, considering indicator functions of intervals, we get We show a fundamental lemma that works as starting point for our results.

Lemma 2.2. Consider an interval
Proof. It is obvious that D p,N,α (·) is in L 2 (T), so that we can apply Parseval's identity and, by (2.1), we get since it easily follows that FD p,N,α (0) = 0. Moreover, we have so that we get our initial claim.

The proof of the main result.
The results in this section are inspired by [2], and we follow its notation. For any positive integer q, we define the function Moreover, the sum of a geometric series gives Because of Lemma 2.1, we are able to split {x n } N −1 n=0 in disjoint sets of equidistant points. More precisely, we get where as the parameter ν increases, we consider (exponentially) smaller sets of equidistant points in T, while the parameter d is less significant but nonetheless necessary.
Hence, we can also split the sum in the previous modulus in several parts, namely Although the long expression, we are just reordering and writing explicitly all these sets because each one of them represents a regular polygon lying on the unit circle in the complex plane. Hence, from (3.1), we further get  Proof. It is easy to see that | sin(παk)| ≤ 1 − δ p λ (k). (3.4) Moreover, notice that for a, b ∈ N + such that a ≤ b, we have

5)
Vol. 120 (2023) Distribution of the van der Corput sequences 289 Consider η ∈ N + such that η ≥ λ, and let η j=1 N j p j−1 be the p-adic expansion of N . Starting with (3.2

) and by (3.3)-(3.4)-(3.5), it follows that
With the change of variable τ = η − ν − 1, and since 0 ≤ N j < p for all j, we finally get which is clearly bounded by a constant depending only on p and λ.
It is time to prove the main result of this paper.
Proof of Theorem 1.3. We break the proof in two parts, first we show that the right inequality holds. As before, let η j=1 N j p j−1 (with N η = 0) be the p-adic expansion of N . Let k be a positive integer, we respectively define L(k) and G(k) as the lowest and the greatest exponents (plus 1) appearing in the p-adic expansion of k, namely k = G(k) m=L(k) β m p m−1 with β L(k) = 0 and β G(k) = 0. In the case of L(k) ≤ η, starting with (3.3) and recalling that 0 ≤ N j < p for all j, we get Also, it is a well-known inequality that therefore, starting with (3.2), we get Now, write L p,N,α as an increasing sequence of indices { q } #Lp,N,α q=1 and, for the sake of construction, add 0 = 0 and 1+#Lp,N,α = η. Notice that all the indices (if any) in between two consecutive q are such that the respective coefficients α are either all 0 or all p − 1.
In the case of q < L(k) ≤ G(k) ≤ q+1 , we proceed to show that In fact, suppose without loss of generality that all the such that q < < q+1 are zeros (the other case is similar), so that we get In the case L(k) ≤ q < G(k) ≤ q+1 , we trivially get αk ≤ 1. Then it is useful to define Moreover, notice that #{k ∈ N | L(k) = r, G(k) = s} ≤ p s−r+1 and that Finally, taking into account what we have gathered so far and reordering the sum in (3.7), we get Vol. 120 (2023) Distribution of the van der Corput sequences 291 so, writing explicitly all the sums, it follows that so that our initial claim on the right inequality holds. Now, we proceed to show that the left inequality holds. Notice that for q ∈ L p,N,α it holds and, from Definition 1.2, it follows that ∞ n=1 α q +n−2 p −n ∈ 1 p 2 , and, without loss of generality, we assume θ η ≡ 0 for all η. Also, for any η ∈ N such that η ≡ 0 (mod 4), define then notice that p η−3 ≤ M η < p η−2 , and therefore #L 0 p,Mη,α = #L 0 p,p η−3 ,α . Now, write L 0 p,Mη,α as an increasing sequence of indices { g } #L 0 p,Mη ,α g=1 and finally, starting with (3.2) and by (3.6)-(3.8), we get (3.10) Notice, by construction, that for ν and g such that 0 ≤ η − ν − 1 ≤ g − 2 (or equivalently δ p η−ν−1 (p g −2 ) = 1), we have that the fractional part Finally, starting with (3.3), we get so that, taking the imaginary part, we get (3.12) Recall that the sine function sin(2π·) is increasing and positive in the interval (0, 1 4 ) so, from (3.11), it follows that On the other hand, whenever η − ν − 1 > g − 2, we trivially get δ p η−ν−1 (p g −2 ) = 0.
Hence, letting η → ∞, we obtain that our initial claim holds.
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