Chebyshevskii Sbornik
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Chebyshevskii Sb.: Year: Volume: Issue: Page: Find

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

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)
References: PDF file   HTML file
UDC: 517
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} 

• http://mi.mathnet.ru/eng/cheb479
• http://mi.mathnet.ru/eng/cheb/v17/i2/p56

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
•  Number of views: This page: 164 Full text: 71 References: 18