On squares in special sets of finite fields

A large part of number theory deals with arithmetic properties of numbers with “missing digits” (that is numbers which digits in a number system with a fixed base belong to a given set). The present paper explores the analog of such a similar problem in the finite field.
We consider the linear vector space formed by the elements of the finite field $\mathbb{F}_q$ with $q=p^r$ over $\mathbb{F}_p$. Let $\{a_1,\ldots,a_r\}$ be a basis of this space. Then every element $x\in\mathbb{F}_q$ has a unique representation in the form $\sum_{j=1}^r c_ja_j$ with $c_j\in\mathbb{F}_p$; the coefficients $c_j$ may be called “digits”. Let us fix the set $\mathcal{D}\subset\mathbb{F}_p$ and let $W_{\mathcal{D}}$ be the set of all elements $x\in\mathbb{F}_q$ such that all its digits belong to the set $\mathcal{D}$. In this connection the elements of $\mathbb{F}_p\setminus\mathcal{D}$ may be called “missing digits”. In a recent paper of C.Dartyge, C.Mauduit, A.Sárközy it has been shown that if the set $\mathcal{D}$ is quite large then there are squares in the set $W_{\mathcal{D}}$. In this paper more common problem is considered. Let us fix subsets $D_1,\ldots,D_r\subset\mathbb{F}_p$ and consider the set $W=W(D_1,\ldots,D_r)$ of all elements $x\in\mathbb{F}_q$ such that $c_j\in D_j$ for all $1\leq j \leq r$. We prove an estimate for the number of squares in the set $W$, which implies the following assertions:

• if $\prod\limits_{i=1}^r|D_i| \geq (2r-1)^rp^{r(1/2+\varepsilon)}$ for some $\varepsilon>0$, then the asymptotic formula $|W\cap Q|=$ $=|W|\left(\frac12+O(p^{-\varepsilon/2})\right)$ is valid;
• if $\prod\limits_{i=1}^r |D_i|\geq 8(2r-1)^rp^{r/2}$, then there exist nonzero squares in the set $W$.

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