Hybrid bounds on two-parametric family Weyl sums along smooth curves

Abstract

We obtain a new bound on Weyl sums with degree $k\ge 2$ polynomials of the form $(\tau x+c)\omega (n)+xn$, $n=1,2,\dots$, with fixed $\omega (T)\in \mathbb{Z}[T]$ and $\tau \in \mathbb{R}$, which holds for almost all $c\in [0,1)$ and all $x\in [0,1)$. We improve and generalise some recent results of M.~B.~Erdogan and G.~Shakan (2019), whose work also shows links between this question and some classical partial differential equations. We extend this to more general settings of families of polynomials $xn+y\omega (n)$ for all $(x,y)\in [0,1{)}^2$ with $f(x,y)=z$ for a set of $z\in [0,1)$ of full Lebesgue measure, provided that $f$ is some Hölder function.

18 pages

References in corpus (2)

• On effective equidistribution for higher step nilflows
• Fourier series in BMO with number theoretical implications

Cited by in corpus (1)

• An $L^4$ maximal estimate for quadratic Weyl sums