Diophantine inequalities for generic ternary diagonal forms

TL;DR

This paper investigates Diophantine inequalities involving ternary diagonal forms of degree k, aiming to find solutions of a certain size under conditions on the parameters, extending previous quadratic results to higher degrees.

Contribution

It generalizes Bourgain's work on quadratic forms to forms of arbitrary degree k, analyzing solutions on average over the parameter \

Findings

Established bounds for solutions in the average over lp_3

Extended methods from quadratic to higher-degree forms

Provided new insights into the distribution of solutions for diagonal forms

Abstract

Let k\geq 2 and consider the Diophantine inequality |x_1^k-\alp_2 x_2^k-\alp_3 x_3^k| <\tet. Our goal is to find non-trivial solutions in the variables x_i, 1\leq i\leq 3, all of size about P, assuming that \tet is sufficiently large. We study this problem on average over \alp_3 and generalize previous work of Bourgain on quadratic ternary diagonal forms to general degree k.