Double exponential sums over thin sets

Abstract

We estimate double exponential sums of the form Sa(X, Y) = E E exp(27riaI0xY/p), xGE X yG Y where V is of multiplicative order t modulo the prime p and X and Y are arbitrary subsets of the residue ring modulo t. In the special case t = p 1, our bound is nontrivial for I XI > I Yl > p15/16+6 with any fixed 6 > 0, while if in addition we have Xl > p1-6/4 it is nontrivial for I YI ? p3/4+6. Let p be a prime and let Fp be a finite field of p elements. For an integer m > 1 we denote by Z,m = {0, . .. , m 1} the residue ring modulo m. We also identify Fp with the set {o,... ,p-1}. Finally we define e(z) = exp(27ri/p) and use log z for the natural logarithm of z. Throughout the paper the implied constants in symbols 'O', ' ' may occasionally, where obvious, depend on the small positive parameter E and are absolute otherwise (we recall that A A are equivalent to A 0 (B)). We fix an element ) E IFp of multiplicative order t, that is, 79s 7 1, 1 pl5/16+6 with any fixed 6 > 0. Further examples are given below. Our results rely on the following estimate for certain double exponential sums from [1]; see the proof of Theorem 8 of that paper. Let A E Fp be of multiplicative order T. For any a, b (E IF we have the bound 4 (1) E| e(a\v +bAuv) 1 satisfies (2) T(m) < m(1). 2 Our main estimate is the following. Theorem. For any sets X, Y C Zt, the bound max ISa( X, Y)| << I x 1/21 yI5/6tl/2pl/8+E aEF* p holds. Proof. For a divisor dlt we denote by Y(d) the subset of y Y Y with gcd(y, t) d. Then ISa(X, Y)I ? Z ud| dlt