Growth polynomials for additive quadruples and $(h,k)$-tuples

TL;DR

This paper studies the polynomial growth of the number of solutions to certain additive equations within integer intervals, providing explicit formulas for these growth polynomials.

Contribution

The paper explicitly constructs the growth polynomials for the number of solutions to additive quadruples and $(h,k)$-tuples within integer intervals.

Findings

Growth polynomials are polynomial functions in $n$ for large $n$.

Explicit formulas for these growth polynomials are derived.

Results apply to equations involving sums of elements in integer intervals.

Abstract

Consider the interval of integers $I_{m,n} = \{m, m+1, m+2,\ldots, m+n-1 \}$. For fixed integers $h,k,m$, and $c$, let $\Phi_{h,k,m}^{(c)}(n)$ denote the number of solutions of the equation $(a_1+\cdots + a_h)- (a_{h+1} + \cdots + a_{h+k})=c$ with $a_i \in I_{m,n}$ for all $i=1,\ldots, h+k$. This is a polynomial in $n$ for all sufficiently large $n$, and the growth polynomial is constructed explicitly.