Arbitrarily long gaps between the values of positive-definite cubic and biquadratic diagonal forms

TL;DR

This paper investigates the distribution of values of positive-definite diagonal forms of degrees 3 and 4, establishing bounds on the size of gaps between representable integers and demonstrating the existence of arbitrarily long gaps with specific properties.

Contribution

It proves the existence of arbitrarily long gaps between representable integers for certain forms and provides explicit bounds on the size of these gaps for degrees 3 and 4.

Findings

Existence of arbitrarily long sequences of integers not representable as sums of s nonnegative s-th powers for s=3,4.

Gaps of size O(√log N / (log log N)^2) for s=3.

Gaps of size O(log log log N / log log log log N) for s=4 under certain conditions.

Abstract

For $s=3,4$, we prove the existence of arbitrarily long sequences of consecutive integers none of which is a sum of $s$ nonnegative $s$-th powers. More generally, we study the existence of gaps between the values $\leq N$ of diagonal forms of degree $s$ in $s$ variables with positive integer coefficients. We find: (1) gaps of size $O\left(\frac{\sqrt {\log N}}{(\log \log N)^2}\right)$ when $s=3$; (2) gaps of size $O\left(\frac{\log\log\log N}{\log\log\log\log N}\right)$ if $s=4$ and the form, up to permutation of the variables, is not equal to $a (c_1x_1)^4+b (c_2 x_2)^4+4 a (c_3x_3)^4+4b(c_4x_4)^4$.