# Representation of integers by sparse binary forms

## TL;DR

This paper establishes new upper bounds for solutions to inequalities involving sparse binary forms, showing that solutions are limited based on the form's sparsity and confirming several longstanding conjectures.

## Contribution

It provides the first sharp bounds for solutions to inequalities with sparse binary forms, advancing understanding of their solution structure and confirming previous conjectures.

## Key findings

- Bounds depend on the number of non-zero coefficients
- Sharp linear bounds for highly sparse forms
- Affirmative answers to conjectures by Mueller and Schmidt

## Abstract

We will give new upper bounds for the number of solutions to the inequalities of the shape $|F(x , y)| \leq h$, where $F(x , y)$ is a sparse binary form, with integer coefficients, and $h$ is a sufficiently small integer in terms of the absolute value of the discriminant of the binary form $F$. Our bounds depend on the number of non-vanishing coefficients of $F(x , y)$. When $F$ is really sparse, we establish a sharp upper bound for the number of solutions that is linear in terms of the number of non-vanishing coefficients. This work will provide affirmative answers to a number of conjectures posed by Mueller and Schmidt in 1988, for special but important cases.