Balance and pattern distribution of sequences derived from pseudorandom subsets of $\mathbb{Z}_q$

Abstract

Let $q$ be a positive integer and $\mathcal{S}=\left\{x_0,x_1,\ldots,x_{T-1}\right\}\subseteq\mathbb{Z}_q=\{0,1,\ldots,q-1\}$ with $$0\leq x_0<x_1<\ldots< x_{T-1}\leq q-1.$$ We derive from $\mathcal{S}$ three (finite) sequences. 1. For an integer $M\geq 2$ let $(s_n)$ be the $M$-ary sequence defined by \begin{eqnarray*} s_n\equiv x_{n+1}-x_n \bmod M, \qquad n=0,1,\ldots, T-2. \end{eqnarray*} 2. For an integer $m\geq 2$ let $(t_n)$ be the binary sequence defined by \begin{eqnarray*} t_n=\left\{\begin{array}{ll} 1, & \hbox{if } 1\leq x_{n+1}-x_n\leq m-1, \\ 0, & \hbox{otherwise}, \end{array}\right. \qquad n=0,1,\ldots, T-2. \end{eqnarray*} 3. Let $(u_n)$ be the characteristic sequence of $\mathcal{S}$, \begin{eqnarray*} u_n=\left\{\begin{array}{ll} 1, & \hbox{if } n\in \mathcal{S}, \\ 0, & \hbox{otherwise}, \end{array}\right. \qquad n=0,1,\ldots, q-1. \end{eqnarray*} We study the balance and pattern distribution of the sequences $(s_n)$, $(t_n)$ and $(u_n)$. For sets $\mathcal{S}$ with desirable pseudorandom properties, more precisely, sets with low correlation measures, we show the following: 1. The sequence $(s_n)$ is (asymptotically) balanced and has uniform pattern distribution if $T$ is of smaller order of magnitude than $q$. 2. The sequence $(t_n)$ is balanced and has uniform pattern distribution if $T$ is approximately $\left(1-\frac{1}{2^{1/(m-1)}}\right)q$. 3. The sequence $(u_n)$ is balanced and has uniform pattern distribution if $T$ is approximately $\frac{q}{2}$. These results are motivated by earlier results for the sets of quadratic residues and primitive roots modulo a prime. We unify these results and derive many further (asymptotically) balanced sequences with uniform pattern distribution from pseudorandom subsets.