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In this paper, we show that for a twice differentiable function g having countable zeros and for Lebesgue almost every β > 1 , the sequence (e 2 π i β n g (β) ) n ∈ N is orthogonal to all topological dynamical systems of zero entropy. To this end, we define the Chowla property and the Sarnak property for numerical sequences taking values 0 or complex numbers of modulus 1. We prove that the Chowla property implies the Sarnak property and show that for Lebesgue almost every β > 1 , the sequence (e 2 π i β n ) n ∈ N shares the Chowla property. It is also discussed whether the samples of a given random sequence have the Chowla property almost surely. Some dependent random sequences having almost surely the Chowla property are constructed. [ABSTRACT FROM AUTHOR]

We study the problem of constructing sequences (x n) n = 1 ∞ on [0, 1] in such a way that D N ∗ = sup 0 ≤ x ≤ 1 # 1 ≤ i ≤ N : x i ≤ x N - x is small. A result of Schmidt shows that for all sequences sequences (x n) n = 1 ∞ on [0, 1] we have D N ∗ ≳ (log N) N - 1 for infinitely many N, several classical constructions attain this growth. We describe a type of uniformly distributed sequence that seems to be completely novel: given x 1 , ⋯ , x N - 1 , we construct x N in a greedy manner x N = arg min min k | x - x k | ≥ N - 10 ∑ k = 1 N - 1 1 - log (2 sin (π | x - x k |)). We prove that D N ∗ ≲ (log N) N - 1 / 2 and conjecture that D N ∗ ≲ (log N) N - 1 . Numerical examples illustrate this conjecture in a very impressive manner. We also establish a discrepancy bound D N ∗ ≲ (log N) d N - 1 / 2 for an analogous construction in higher dimensions and conjecture it to be D N ∗ ≲ (log N) d N - 1 . [ABSTRACT FROM AUTHOR]

The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, in the one-dimensional setting, the pair correlation statistics measures the distribution of spacings between sequence elements in the unit interval at distances of order of the mean spacing 1 / N. In the d-dimensional case, of course, the order of the mean spacing is 1 / N 1 d , and—in our concept—the distance of sequence elements will be measured by the supremum-norm. Additionally, we show that, in some sense, almost all sequences satisfy this new concept and we examine the link to uniform distribution. The metrical pair correlation theory is investigated and it is proven that a class of typical low-discrepancy sequences in the high-dimensional unit cube do not have Poissonian pair correlations, which fits the existing results in the one-dimensional case. [ABSTRACT FROM AUTHOR]

Let $$p$$ be a prime. We study the distribution of points on a class of curves $$C$$ over $$\mathbb{F }_p$$ inside very small rectangles $$\mathcal{B }$$ for which the Weil bound fails to give nontrivial information. In particular, we show that the distribution of points on $$C$$ over long rectangles is Gaussian. [ABSTRACT FROM AUTHOR]

In this paper we study uniform distribution properties of digital sequences over a finite field of prime order. In 1998 it was shown by Larcher that for almost all $$s$$ -dimensional digital sequences $$\mathcal{S }$$ the star discrepancy $$D_N^*$$ satisfies an upper bound of the form $$D_N^*(\mathcal{S })=O((\log N)^s (\log \log N)^{2+\varepsilon })$$ for any $$\varepsilon >0$$ . Generally speaking it is much more difficult to obtain good lower bounds for specific sequences than upper bounds. Here we show that Larchers result is best possible up to some $$\log \log N$$ term. More detailed, we prove that for almost all $$s$$ -dimensional digital sequences $$\mathcal{S }$$ the star discrepancy satisfies $$D_N^*(\mathcal{S }) \ge c(q,s) (\log N)^s \log \log N$$ for infinitely many $$N \in \mathbb{N }$$ , where $$c(q,s)>0$$ only depends on $$q$$ and $$s$$ but not on $$N$$ . [ABSTRACT FROM AUTHOR]

Rényi (Mat Lapok 7:77-100, 1956) made a definition that gives a generalization of simple normality in the context of Q-Cantor series. In Mance ), a definition of Q-normality was given that generalizes the notion of normality in the context of Q-Cantor series. In this work, we examine both Q-normality and Q-distribution normality, treated in Lafer (Normal numbers with respect to Cantor series representation, 1974) and S̆alát (Czechoslovak Math J 18(93):476-488, 1968). Specifically, while the non-equivalence of these two notions is implicit in Lafer (Normal numbers with respect to Cantor series representation. Washington State University, 1974), in this paper, we give an explicit construction witnessing the nontrivial direction. That is, we construct a base Q as well as a real x that is Q-normal yet not Q-distribution normal. We next approach the topic of simultaneous normality by constructing an explicit example of a base Q as well as a real x that is both Q-normal and Q-distribution normal. [ABSTRACT FROM AUTHOR]

Given and , we define by setting if and only if , where denotes the fractional part of α, i.e. α is considered as an element of the torus . If the topological boundary of A has Haar measure 0, then is called a Hartman sequence, which is a generalisation of Kronecker and Beatty sequences. In this article we answer a question of Winkler by showing explicitly for which sets , and vectors , we have . The main tool of the proof is Weyl’s theorem on uniform distribution. [ABSTRACT FROM AUTHOR]

We introduce several concepts of discrepancy for sequences on the Sierpiński gasket. Furthermore a law of iterated logarithm for the discrepancy of trajectories of Brownian motion is proved. The main tools for this result are regularity properties of the heat kernel on the Sierpiński gasket. Some of the results can be generalized to arbitrary nested fractals in the sense of T. Lindstrøm. [ABSTRACT FROM AUTHOR]