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Chebyshevskii Sb., 2016, Volume 17, Issue 2, Pages 56–63 (Mi cheb479)  

This article is cited in 1 scientific paper (total in 1 paper)

On squares in special sets of finite fields

M. R. Gabdullinab

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 $\prod\limits_{i=1}^r|D_i| \geq (2r-1)^rp^{r(1/2+\varepsilon)}$ for some $\varepsilon>0$, then the asymptotic formula $|W\cap Q|=$ $=|W|(\frac12+O(p^{-\varepsilon/2}))$ is valid;
  • if $\prod\limits_{i=1}^r |D_i|\geq 8(2r-1)^rp^{r/2}$, 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 14-11-00702
The work is supported by the grant from the Russian Science Foundation (Project 14-11-00702).


Full text: PDF file (579 kB)
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UDC: 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 56--63
\mathnet{http://mi.mathnet.ru/cheb479}
\elib{https://elibrary.ru/item.asp?id=26254424}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. 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  mathnet  crossref  crossref  mathscinet  isi  elib
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