Combinational properties of sets of residues modulo a prime and the Erdos-Graham problem

Consider an arbitrary epsilon > 0 and a sufficiently large prime p > 2. It is proved that, for any integer a, there exist pairwise distinct integers x(1), x(2), x(N), where N = 8([1/epsilon + 1/2] + 1)(2) such that 1 <= x(i) <= p(epsilon), i = 1, N, and a equivalent to x(1)(-1) + center dot center dot center dot + x(N)(-1) (mod p), where x(i)(-1) is the least positive integer satisfying x(i)(-1)x(i) equivalent to 1 (mod p). This improves a result of Sparlinski.