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Register a new userMetric theory of lower bounds on Weyl sums
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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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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.
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).
For a subset A of a finite abelian group G we define Sigma(A)={sum_{ain B}a:Bsubset A}. In the case that Sigma(A) has trivial stabiliser, one may deduce that the size of Sigma(A) is at least quadratic in |A|; the bound |Sigma(A)|>= |A|^{2}/64 has recently been obtained by De Vos, Goddyn, Mohar and Samal. We improve this bound to the asymptotically best possible result |Sigma(A)|>= (1/4-o(1))|A|^{2}. We also study a related problem in which A is any subset of Z_{n} with all elements of A coprime to n; it has recently been shown, by Vu, that if such a set A has the property Sigma(A) is not Z_{n} then |A|=O(sqrt{n}). This bound was improved to |A|<= 8sqrt{n} by De Vos, Goddyn, Mohar and Samal, we further improve the bound to the asymptotically best possible result |A|<= (2+o(1))sqrt{n}.
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 obtain the exact value of the Hausdorff dimension of the set of coefficients of Gauss sums which for a given $alpha in (1/2,1)$ achieve the order at least $N^{alpha}$ for infinitely many sum lengths $N$. For Weyl sums with polynomials of degree $dge 3$ we obtain a new upper bound on the Hausdorff dimension of the set of polynomial coefficients corresponding to large values of Weyl sums. Our methods also work for monomial sums, match the previously known lower bounds, just giving exact value for the corresponding Hausdorff dimension when $alpha$ is close to $1$. We also obtain a nearly tight bound in a similar question with arbitrary integer sequences of polynomial growth.
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