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

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 $a_1,\dots ,a_r$ be a basis of this space. Then the elements $x$ of ${\mathbb{F}}_q$ have a unique representation in the form ${\sum }_{j=1}^r{c}_j{a}_j$ 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 $j=1,\dots ,r$. 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.Sárközy.

in Russian