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Register a new userBounds on the Norms of Maximal Operators on Weyl Sums
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We obtain new estimates on the maximal operator applied to the Weyl sums. We also consider the quadratic case (that is, Gauss sums) in more details. In wide ranges of parameters our estimates are optimal and match lower bounds. Our approach is based on a combination of ideas of Baker (2021) and Chen and Shparlinski (2020).
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We prove that the Hausdorff dimension of the set $mathbf{x}in [0,1)^d$, such that $$ left|sum_{n=1}^N expleft(2 pi ileft(x_1n+ldots+x_d n^dright)right) right|ge c N^{1/2} $$ 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 $dge 3$. We also obtain similar bounds for the Hausdorff dimension of the set of large sums with monomials $xn^d$.
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Shifted convolution sums play a prominent role in analytic number theory. We investigate pointwise bounds, mean-square bounds, and average bounds for shifted convolution sums for Hecke eigenforms.
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 $$ cN^{1/2}le left|sum^N_{n=1}expleft (2 pi i left (x_1n+ldots+x_d n^dright)right) right|le C N^{1/2}$$ 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 $d eq 4$. Finally, we obtain lower bounds for the Hausdorff dimension of large values of general exponential polynomials.
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