Waring's problem with shifts

TL;DR

This paper extends Waring's problem to sums of shifted powers with irrational shifts, establishing bounds on the number of variables needed for approximation and providing asymptotic formulas for sufficiently many variables.

Contribution

It introduces a real analogue of Waring's problem involving shifted powers and derives bounds and asymptotic formulas for sums of such powers with irrational shifts.

Findings

Bound on the number of variables for approximation when $k \\ge 4$

Asymptotic formula for sums with at least $2k^2 - 2k + 3$ variables

Results for sums of general univariate degree $k$ polynomials

Abstract

Let $\mu_1, \ldots, \mu_s$ be real numbers, with $\mu_1$ irrational. We investigate sums of shifted $k$th powers $\mathfrak{F}(x_1, \ldots, x_s) = (x_1 - \mu_1)^k + \ldots + (x_s - \mu_s)^k$. For $k \ge 4$, we bound the number of variables needed to ensure that if $\eta$ is real and $\tau > 0$ is sufficiently large then there exist integers $x_1 > \mu_1, \ldots, x_s > \mu_s$ such that $|\mathfrak{F}(\mathbf{x}) - \tau| < \eta$. This is a real analogue to Waring's problem. When $s \ge 2k^2-2k+3$, we provide an asymptotic formula. We prove similar results for sums of general univariate degree $k$ polynomials.