On squares in subsets of finite fields with restrictions on coefficients of basis decomposition

TL;DR

This paper estimates the number of squares in subsets of finite fields with restricted coefficients in a basis, providing weaker conditions for the existence of squares than previous results.

Contribution

It introduces new bounds on the count of squares in structured subsets of finite fields with coefficient restrictions, improving upon prior conditions for their existence.

Findings

Derived estimates for the number of squares in subsets with coefficient restrictions.

Established weaker sufficient conditions for the existence of squares in these subsets.

Compared results to previous work, showing improved bounds.

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 $D$ be a subset of $\mathbb{F}_p$. We consider the set $W_D$ of elements of $\mathbb{F}_q$ such that $c_j\in D$ for all $j=1,\ldots,r$. We give an estimate for the number of squares in the set $W_D$ which implies a weaker sufficient condition for the existence of squares in the set $W_D$ than in the recent paper of C.Dartyge, C.Mauduit, A.S\'ark\"ozy.