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On squares in special sets of finite fields
M. R. Gabdullin^{ab} ^{a} Lomonosov Moscow State University
^{b} Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
Abstract:
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 $\prod\limits_{i=1}^rD_i \geq (2r1)^rp^{r(1/2+\varepsilon)}$ for some $\varepsilon>0$, then the asymptotic formula $W\cap Q=$ $=W(\frac12+O(p^{\varepsilon/2}))$ is valid;
 if $\prod\limits_{i=1}^r D_i\geq 8(2r1)^rp^{r/2}$, then there exist nonzero squares in the set $W$.
Bibliography: 18 titles.
Keywords:
finite fields, squares, character sums.
Funding Agency 
Grant Number 
Russian Science Foundation 
141100702 
The work is supported by the grant from the Russian Science Foundation (Project 141100702). 
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517 Received: 05.01.2016 Revised: 10.06.2016
Citation:
M. R. Gabdullin, “On squares in special sets of finite fields”, Chebyshevskii Sb., 17:2 (2016), 56–63
Citation in format AMSBIB
\Bibitem{Gab16}
\by M.~R.~Gabdullin
\paper On squares in special sets of finite fields
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 2
\pages 5663
\mathnet{http://mi.mathnet.ru/cheb479}
\elib{https://elibrary.ru/item.asp?id=26254424}
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This publication is cited in the following articles:

M. R. Gabdullin, “On the Squares in the Set of Elements of a Finite Field with Constraints on the Coefficients of Its Basis Expansion”, Math. Notes, 101:2 (2017), 234–249

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