On squares in special sets of finite fields

M. R. Gabdullin

Čebyševskij sbornik, Tome 17 (2016) no. 2, pp. 56-63.

Résumé

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 $W$. Bibliography: 18 titles.

Détail

Keywords: finite fields, squares, character sums.

Bibliographie

[1] Banks W. D., Conflitti A., Shparlinski I. E., “Character sums over integers with restricted $g$-ary digits”, Illinois J. Math., 46:3 (2002), 819–836 | MR | Zbl

[2] Banks W. D., Shparlinski I. E., “Arithmetic properties of numbers with restricted digits”, Acta Arith., 112 (2004), 313–332 | DOI | MR | Zbl

[3] Col S., Propriétés multiplicatives d'entiers soumis à des contraintes digitales, Thèse de doctorat de mathématiques de l'Université Henri Poincaré-Nancy. Vol. 1, 2006

[4] Col S., “Diviseurs des nombres ellipséphiques”, Periodica Mathematica Hungarica, 58:1 (2009), 1–23 | DOI | MR | Zbl

[5] Coquet J., “On the uniform distribution modulo one of some subsequences of polynomial sequences”, J. Number Theory, 10:3 (1978), 291–296 | DOI | MR | Zbl

[6] Coquet J., “On the uniform distribution modulo one of some subsequences of polynomial sequences”, J. Number Theory, 12:2 (1980), 244–250 | DOI | MR | Zbl

[7] Coquet J., “Graphes connexes, représentation de entiers et équirépartition”, J. Number Theory, 16:3 (1983), 363–375 | DOI | MR | Zbl

[8] Dartyge C., Mauduit C., “Nombres presque premiers dont l`écriture en base $r$ ne comporte pas certain chiffres”, Journal of Number Theory, 81 (2000), 270–291 | DOI | MR | Zbl

[9] Dartyge C., Mauduit C., “Ensembles de densité nulle contenant des entiers possédant au plus deux facteurs premiers”, Journal of Number Theory, 91 (2001), 230–255 | DOI | MR | Zbl

[10] Drmota M., Mauduit C., “Weyl sums over integers with affine digits restriction”, Journal of Number Theory, 30 (2010), 2404–2427 | DOI | MR

[11] Erdős P., Mauduit C., Sárközy A., “On the arithmetic properties of integers with missing digits I: Distribution in residue classes”, Journal of Number Theory, 70:2 (1998), 99–120 | DOI | MR | Zbl

[12] Konyagin S. V., Mauduit C., Sárközy A., “On the number of prime factors of integers characterized by digits properties”, Period. Math. Hung., 40 (2000), 37–52 | DOI | MR | Zbl

[13] Dartyge C., Sárközy A., “The sum of digits function in the finite field”, Proc. Amer. Math. Soc., 141:12 (2013), 4119–4124 | DOI | MR | Zbl

[14] Dartyge C., Mauduit C., Sárközy A., “Polynomial values and generators with missing digits in finite fields”, Functiones et Approximatio, 52:1 (2015), 65–74 | DOI | MR | Zbl

[15] Gabdullin M. R., On squares in subsets of finite fields with restrictions on coefficients of basis decomposition (Russian)

[16] Dietmann R., Elsholtz C., Shparlinski I. E., Prescribing the binary digits of squarefree numbers and quadratic residues, arXiv: 1601.04754v1

[17] Wan D., “Generators and irreducible polynomials over finite fields”, Math. Comp., 66 (1997), 1195–1212 | DOI | MR | Zbl

[18] Winterhof A., “Characters sums, primitive elements, and powers in finite fields”, Journal of Number Theory, 91 (2001), 153–161 | DOI | MR