  A special case of the Menshov--Rademacher theorem implies for almost all
polynomials $x_1Z+\ldots +x_d Z^{d} \in {\mathbb R}[Z]$ of degree $d$ for the
Weyl sums satisfy the upper bound $$ \left| \sum_{n=1}^{N}\exp\left(2\pi i
\left(x_1 n+\ldots +x_d n^{d}\right)\right) \right| \leqslant N^{1/2+o(1)},
\qquad N\to \infty. $$ Here we investigate the exceptional sets of coefficients
$(x_1, \ldots, x_d)$ with large values of Weyl sums for infinitely many $N$,
and show that in terms of the Baire categories and Hausdorff dimension they are
quite massive, in particular of positive Hausdorff dimension in any fixed cube
inside of $[0,1]^d$. We also use a different technique to give similar results
for sums with just one monomial $xn^d$. We apply these results to show that the
set of poorly distributed modulo one polynomials is rather massive as well.

Number Theory