Sums of Gal and applications

We evaluate the asymptotic size of various sums of Gal type, in particularS(M):= n-ary sumation m,n is an element of M(m,n)[m,n],where M is a finite set of integers. Elaborating on methods recently developed by Bondarenko and Seip, we obtain an asymptotic formula for log (sup(vertical bar M vertical bar=N)S(M)/N) and derive new lower bounds for localized extreme values of the Riemann zeta-function, for extremal values of some Dirichlet L-functions at s=12, and for large character sums.