  We obtain the exact value of the Hausdorff dimension of the set of
coefficients of Gauss sums which for a given $\alpha \in (1/2,1)$ achieve the
order at least $N^{\alpha}$ for infinitely many sum lengths $N$. For Weyl sums
with polynomials of degree $d\ge 3$ we obtain a new upper bound on the
Hausdorff dimension of the set of polynomial coefficients corresponding to
large values of Weyl sums. Our methods also work for monomial sums, match the
previously known lower bounds, just giving exact value for the corresponding
Hausdorff dimension when $\alpha$ is close to $1$. We also obtain a nearly
tight bound in a similar question with arbitrary integer sequences of
polynomial growth.
