A note on polynomial sequences modulo integers

TL;DR

This paper investigates the distribution of polynomial sequences modulo integers, establishing conditions under which they are uniformly distributed or intersect all congruence classes, based on the irrationality of polynomial coefficients.

Contribution

It provides new criteria linking the irrationality of polynomial coefficients to the uniform distribution and congruence class intersection of their integer floor sequences.

Findings

Sequence is uniformly distributed iff polynomial has an irrational coefficient (excluding constant)

Sequence intersects all congruence classes iff polynomial has an irrational coefficient (excluding constant) for even degree

Results apply to nonlinear polynomials with real coefficients

Abstract

We study the uniform distribution of the polynomial sequence $\lambda(P)=(\lfloor P(k) \rfloor )_{k\geq 1}$ modulo integers, where $P(x)$ is a polynomial with real coefficients. In the nonlinear case, we show that $\lambda(P)$ is uniformly distributed in $\mathbb{Z}$ if and only if $P(x)$ has at least one irrational coefficient other than the constant term. In the case of even degree, we prove a stronger result: $\lambda(P)$ intersects every congruence class modulo every integer if and only if $P(x)$ has at least one irrational coefficient other than the constant term.