On squares in special sets of finite fields

TL;DR

This paper investigates the distribution of squares within specific subsets of finite fields, providing estimates and conditions for the presence of squares based on the structure of these subsets.

Contribution

It introduces a new estimate for counting squares in structured subsets of finite fields and establishes conditions for their existence.

Findings

Provides an asymptotic formula for the number of squares in large subsets

Derives sufficient conditions for the existence of squares in these sets

Offers bounds on the number of squares based on subset sizes

Abstract

We consider the linear vector space formed by the elements of the finite fields $\mathbb{F}_q$ with $q=p^r$ over $\mathbb{F}_p$. Let $\{a_1,\ldots,a_r\}$ be a basis of this space. Then the elements $x$ of $\mathbb{F}_q$ have a unique representation in the form $\sum_{j=1}^r c_ja_j$ with $c_j\in\mathbb{F}_p$. Let $D_1,\ldots,D_r$ be subsets of $\mathbb{F}_p$. We consider the set $W=W(D_1,\ldots,D_r)$ of elements of $\mathbb{F}_q$ such that $c_j \in D_j$ for all $j=1,\ldots,r$. We give an estimate for the number of squares in the set $W$ which implies an asymptotic formula for this quantity in the case when the sets $D_1,\ldots,D_r$ are "large on average" and a sufficient condition for the existence of squares in the set $W$.