ACTA ARITHMETICA 105.4 (2002) Some doubly exponential sums over Z m by John B. Friedlander (Toronto), Sergei Konyagin (Moscow) and Igor E. Shparlinski (Sydney) 1. Introduction. For an integer m we denote by Z m the residue ring modulo m and by U m = Z ∗ m the group of units of Z m . Let ϑ ∈ Z m , gcd(ϑ, m) = 1. We recall that the multiplicative order ord m ϑ of an integer ϑ modulo an integer m ≥ 1 with gcd(ϑ, m) = 1 is the smallest positive integer t for which ϑ t ≡ 1 (mod m). Define e d (z ) = exp(2πiz/d). Given an integer ϑ with multiplicative order ord m ϑ = t, for integers a, b, c we define the exponential sum S a,b,c (m, t)= t  x,y=1 e m (aϑ x + bϑ y + cϑ xy ). We obtain a non-trivial upper bound for these sums. Specifically we prove S a,b,c (m, t)= O(t 21/16 m 5/8+ε ) provided that gcd(ac, m) = 1 with a somewhat weaker result for the general case. From this we deduce the uniformity of distribution modulo m of the triples (ϑ x ,ϑ y ,ϑ xy ), x, y =1,...,t, provided that t ≥ m 10/11+ε . As in [2, 3] we actually study the slightly simpler sums W a,c (m, t)= t  y=1    t  x=1 e m (aϑ x + cϑ xy )    for which obviously |S a,b,c |≤ min{W a,c ,W b,c }. 2000 Mathematics Subject Classification : 11L07, 11K45, 11Y16. Research of J. B. Friedlander supported in part by NSERC grant A5123. Research of S. Konyagin supported in part by RFBR grants 02-01-00248 and 00-15- 96109. Research of I. E. Shparlinski supported in part by ARC grant A69700294. [349]