Hybrid bounds on two-parametric family Weyl sums along smooth curves
Abstract
We obtain a new bound on Weyl sums with degree k 2 polynomials of the form (τ x+c) ω(n)+xn, n=1, 2, …, with fixed ω(T) ∈ Z[T] and τ ∈ R, which holds for almost all c∈ [0,1) and all x∈ [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 ω(n) for all (x,y)∈ [0,1)2 with f(x,y)=z for a set of z ∈ [0,1) of full Lebesgue measure, provided that f is some H\"older function.
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