## Large families of subsets arising from Woods problem and their pseudorandomness

Speaker: 刘华宁（西北大学）

Venue: B1044

Time: 2018年5月18日 15:00-16:00

Title: Large families of subsets arising from Woods problem and their pseudorandomness

Abstract: Let $q>2$ be an integer. For any integer $a$ with $(a,q)=1$, there exists unique integer $\overline{a}$ such that $1\leq \overline{a}\leq q$ and $a\overline{a}\equiv 1 \ (\bmod\ q)$. Let $\phi(q)$ be the Euler function, and let $\delta$ be a real constant with $0<\delta\leq 1$. In 1994 A. C. Woods asked whether the limit $$\lim_{n\rightarrow\infty}\frac{\displaystyle\left|\left\{a\in\mathbb{Z}: 1\leq a \leq q, (a,q)=1, |a-\overline{a}|<\delta q\right\}\right|}{\phi(q)}$$ exists as $q\rightarrow\infty$? Many authors have studied the problem and related. In this talk we introduce large families of subsets arising from Woods problem and study their cardinalities. Estimates of character sums over the subsets are given. We study the pesudorandom properties of the subsets and show that their well-distribution measures are very high.