# Robert Tichy: Equidistribution and Pseudorandomness

Time: Wed 2024-02-28 14.00 - 14.50

Location: Institut Mittag-Leffler, Seminar Hall Kuskvillan and Zoom

Video link: Meeting ID: 921 756 1880

Participating: Robert Tichy (TU Graz)

Abstract:

In 1997 C. Mauduit and A. Sarkőzy [Acta Arith vol. 82] introduced various measures for pseudo randomness of binary sequences over $$\{−1,+1\}$$. In this talk the focus lies on the correlation measure of order $$s$$. Let $$E_{N}=\{e_{1},\ldots, e_{N}\}$$ be a finite binary sequence, $$M$$ a positive integer and $$\underline{d}= (d_{1}\ldots, d_{s})\in \mathbb{N}^{s}$$ a correlation vector such that $$O \leq d_{1} < d_{2} < \ldots < d_{s} \leq N - M$$. We set
$$$$V (E_{N},M,\underline{d})= \sum^{M}_{n=1}e_{n+d_{1}}e_{n+d_{2}}\ldots e_{n+d_{s}}.$$$$
Then the correlation measure of order $$s$$ of $$E_N$$ is defined as
$$$$C_{s}(E_{N})= \max_{M,\underline{d}}\mid V (E_{N}, M, \underline{d})\mid.$$$$
We construct binary sequences from certain functions $$f(x)$$ of polynomial growth by $$e_{n}= \chi (f(n)),$$, where
$$$$\chi(x)= \begin{cases}+1\; \text{for} & 0 \leq \{x\} < \frac{1}{2}\\ -1\; \text{for} & \frac{1}{2}\leq \{x\} < 1\end{cases},$$$$
where $$\{\cdot\}$$ denotes the fractional part. The approach works for functions $$f(x)= x^{c}$$ ($$c>1$$ not an integer) and for Hardy fields as introduced by Boshernitzan (1994) to equidistribution theory. The main result is an estimate of the form $$C_{s}(E_{N})\ll N^{1-\eta}$$ for some positive constant $$\eta$$. In contrary, we have for $$f(x)= x^{c}$$ with $$0 < c < 1$$
$$$$C_{2}(E_{N})\gg N,$$$$
whereas such sequences are still uniformly distributed modulo $$1$$. The method is based on van der Corput like estimates of exponential sums. I will also give an overview on measures of pseudorandomness and want to address possible connections to pair correlations as considered by Z. Rudnick, P. Sarnak and followers.