Sums of cubes with shifts

TL;DR

This paper studies sums of shifted cubes, proving density and approximation results for sums involving real shifts, extending classical problems like Waring's problem to shifted variables.

Contribution

It establishes new density and approximation theorems for sums of shifted cubes with real shifts, including asymptotic formulas and generalizations to polynomial sums.

Findings

Existence of integer solutions approximating real targets for s ≥ 9.

Full density results for s ≥ 5.

Asymptotic formulas for s ≥ 11.

Abstract

Let $\mu_1, \ldots, \mu_s$ be real numbers, with $\mu_1$ irrational. We investigate sums of shifted cubes $F(x_1,\ldots,x_s) = (x_1 - \mu_1)^3 + \ldots + (x_s - \mu_s)^3$. We show that if $\eta$ is real, $\tau >0$ is sufficiently large, and $s \ge 9$, then there exist integers $x_1 > \mu_1, \ldots, x_s > \mu_s$ such that $|F(\mathbf{x})- \tau| < \eta$. This is a real analogue to Waring's problem. We then prove a full density result of the same flavour for $s \ge 5$. For $s \ge 11$, we provide an asymptotic formula. If $s \ge 6$ then $F(\mathbf{Z}^s)$ is dense on the reals. Given nine variables, we can generalise this to sums of univariate cubic polynomials.