Let F-P be the field of a prime order p and F-p* be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the bound vertical bar A - A vertical bar + vertical bar AA vertical bar vertical bar A vertical bar(13/12)(log vertical bar A vertical bar)(-4/11) for any subset A subset of F-P with 1 < vertical bar A vertical bar < p(12/23). Then we apply our estimate to obtain explicit bounds for some exponential sums in F-p. We show that for any subsets X, Y, Z subset of F-p* and any complex numbers alpha(x), beta(y), gamma(z) with vertical bar alpha(x)vertical bar <= 1, vertical bar beta(y)vertical bar <= 1, vertical bar gamma(z)vertical bar <= 1, the following bound holds: vertical bar Sigma(x is an element of X) Sigma(y is an element of Y)Sigma(z is an element of Z) alpha(x)beta(y)gamma e(p) (xyz)vertical bar < (vertical bar X vertical bar vertical bar Y vertical bar vertical bar Z vertical bar)(13/16)p(5/8+o(1)). We apply this bound further to show that if H is a subgroup of F-p* with vertical bar H vertical bar > p(1/4), then max((a,p)=1)vertical bar Sigma(x is an element of H)e(p)(ax)vertical bar < vertical bar H vertical bar(9437009/9437184+o(1)). Finally we show that if g is a generator of F-p*, then for any M < p the number of solutions of the equation g(x) + g(y) = g(z) + g(t), 1 <= x, y, z, t <= M is less than M3-1/24+o(1) (1 + (M-2/p)(1/24)). This implies that if p(1/2) < M < p, then max((a,p)=1)vertical bar Sigma(x <= M) e(p)(ag(x))vertical bar < M215/217+o(1).
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