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Mathematics, Number Theory
Abstract:
We prove that there exist positive constants $C$ and $c$ such that for any
integer $d \ge 2$ the set of ${\mathbf x}\in [0,1)^d$ satisfying $$ cN^{1/2}\le
\left|\sum^N_{n=1}\exp\left (2 \pi i \left (x_1n+\ldots+x_d n^d\right)\right)
\right|\le C N^{1/2}$$ for infinitely many natural numbers $N$ is of full
Lebesque measure. This substantially improves the previous results where
similar sets have been measured in terms of the Hausdorff dimension. We also
obtain similar bounds for exponential sums with monomials $xn^d$ when $d\neq
4$. Finally, we obtain lower bounds for the Hausdorff dimension of large values
of general exponential polynomials.
