Hybrid Bounds on Two-Parametric Families of Weyl Sums Along Smooth Curves

We obtain a new bound on Weyl sums with degree k >= 2 polynomials of the form (tau x + c)omega (n) + xn, n = 1, 2, ..., with fixed omega(T) is an element of Z[T] and tau is an element of R, which holds for almost all c E [0, 1) and all x is an element of [0, 1). We improve and generalize some recent results of Erdogan and Shakan (2019), whose work also shows links between this ques-tion 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) is an element of [0, 1)(2) with f (x, y) = z fora set of z is an element of [0, 1) of full Lebesgue measure, provided that f is a Holder function.