**Speaker: **Liu Huaning, Professor, School of Mathematics, Northwest University， China

**Date: **May 18, 2018

**Time**: 3:00 p.m. – 4:00 p.m.

**Location: **1044 Lecture Hall, Block B, Zhixin Building, Central Campus

**Sponsor: **the School of Mathematics

**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.

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Edited by Xing Chenyang, Song Yijun