Metric theory of lower bounds on Weyl sums

We prove that the Hausdorff dimension of the set $\mathbf{x}\in [0,1{)}^d$, such that $$\mid \sum _{n=1}^N\exp \left(2\pi i(x_{1}n+\dots +x_d{n}^d)\right)\mid \ge c{N}^{1\mathrm{/}2}$$ holds for infinitely many natural numbers $N$, is at least $d-1\mathrm{/}2d$ for $d\ge 3$ and at least $3\mathrm{/}2$ 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 $x{n}^d$.