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 ${a_1,\ldots,a_r}$ 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 paper of C.Dartyge, C.Mauduit, A.S\'ark\"ozy.