  We prove that the Hausdorff dimension of the set $\mathbf{x}\in [0,1)^d$,
such that $$ \left|\sum_{n=1}^N \exp\left(2 \pi i\left(x_1n+\ldots+x_d
n^d\right)\right) \right|\ge c N^{1/2} $$ holds for infinitely many natural
numbers $N$, is at least $d-1/2d$ for $d \ge 3$ and at least $3/2$ for $d=2$,
where $c$ is a constant depending only on $d$. This improves the previous lower
bound of the first and third authors for $d\ge 3$. We also obtain similar
bounds for the Hausdorff dimension of the set of large sums with monomials
$xn^d$.
