Gaps between quadratic forms

2025/05/29 by Iyer, Siddharth
#11B05 (Secondary) #11B25 #11B34 #11N56 (Primary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.23428

Abstract

Let \triangle denote the integers represented by the quadratic form x²+xy+y² and \square₂ denote the numbers represented as a sum of two squares. For a non-zero integer a, let S(\triangle,\square₂,a) be the set of integers n such that n ∈ \triangle, and n + a ∈ \square₂. We conduct a census of S(\triangle,\square₂,a) in short intervals by showing that there exists a constant H_a > 0 with # S(\triangle,\square₂,a)∩ [x,x+H_a⋅ x^5/6⋅ log¹⁹x] ≥ x^5/6-ε for large x. To derive this result and its generalization, we utilize a theorem of Tolev (2012) on sums of two squares in arithmetic progressions and analyse the behavior of a multiplicative function found in Blomer, Brüdern & Dietmann (2009). Our work extends a classical result of Estermann (1932) and builds upon work of Müller (1989).

Citations

• The additive problem for the number of representations as a sum of two squares
• On the remainder term in the circle problem in an arithmetic progression

Related

• Longer gaps between values of binary quadratic forms
• On pairs of triangular numbers whose product is a perfect square and pairs of intervals of successive integers with equal sums of squares
• Sums of four squares with a certain restriction
• Sums of two squares in short intervals
• Diophantine approximation with sums of two squares II