ON SQUARES IN SPECIAL SETS OF FINITE FIELDS

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 Fq with q = p r over Fp. Let {a1, . . . , ar} be a basis of this space. Then every element x ∈ Fq has a unique representation in the form Pr j=1 cjaj with cj ∈ Fp; the coefficients cj may be called “digits”. Let us fix the set D ⊂ Fp and let WD be the set of all elements x ∈ Fq such that all its digits belong to the set D. In this connection the elements of Fp \ D may be called “missing digits”. In a recent paper of C.Dartyge, C.Mauduit, A.S´ark¨ozy it has been shown that if the set D is quite large then there are squares in the set WD. In this paper more common problem is considered.

Let us fix subsets D1, . . . , Dr ⊂ Fp and consider the set W = W(D1, . . . , Dr) of all elements x ∈ Fq such that cj ∈ Dj for all 1 ≤ j ≤ r. We prove an estimate for the number of squares in the set W, which implies the following assertions:

1) if Qr i=1 |Di | ≥ (2r − 1)rp r(1/2+ε) for some ε > 0, then the asymptotic formula |W ∩ Q| = = |W| 1 2 + O(p −ε/2 ) is valid;

2) if Qr i=1 |Di | ≥ 8(2r − 1)rp r/2 , then there exist nonzero squares in the set W.