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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 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 g
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 $d
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
A $k$-sum of a set $Asubseteq mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $kwedge A$ for the set of $k$-sums of $
We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results of Flajol