Higher Order Oscillation and Uniform Distribution

TL;DR

This paper introduces a new class of higher order oscillating sequences involving exponential functions with polynomial and non-polynomial components, and proves their uniform distribution properties for almost all parameters.

Contribution

It demonstrates the existence of higher order oscillating sequences beyond the Möbius function and establishes their uniform distribution for a broad class of parameters and polynomials.

Findings

Sequences of the form $(e^{2\pi i \alpha eta^{n}g(eta)})$ are higher order oscillating.

Sequences $(ig(\alpha eta^{n}g(eta)+Q(n)ig))$ are uniformly distributed modulo 1.

Results hold for almost all $eta > 1$ and fixed $eta$ with almost all $\alpha$.

Abstract

It is known that the M\"obius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form $(e^{2\pi i \alpha \beta^{n}g(\beta)})_{n\in \N}$, for a non-decreasing twice differentiable function $g$ with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number $\alpha$ and almost all real numbers $\beta >1$ (alternatively, for a fixed real number $\beta >1$ and almost all real numbers $\alpha$) and for all real polynomials $Q(x)$, sequences $\big(\alpha \beta^{n}g(\beta)+Q(n)\big)_{n\in \N}$ are uniformly distributed modulo $1$.