Arithmetic progressions in binary quadratic forms and norm forms

TL;DR

This paper establishes an upper bound on the length of arithmetic progressions represented by binary quadratic forms and norm forms, depending solely on the form and common difference, improving previous results.

Contribution

It provides a new, sharper upper bound for arithmetic progressions in binary quadratic and norm forms, advancing understanding of their structure.

Findings

Upper bound depends only on the form and common difference

Significant improvement over previous bounds for quadratic forms

Applicable to irreducible integral forms

Abstract

We prove an upper bound for the length of an arithmetic progression represented by an irreducible integral binary quadratic form or a norm form, which depends only on the form and the progression's common difference. For quadratic forms, this improves significantly upon an earlier result of Dey and Thangadurai.