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Gaps between quadratic forms

This paper establishes that for any non-zero integer $a$ , the set of integers representable by the quadratic form $x^2+xy+y^2$ whose shift by $a$ is a sum of two squares contains a significant number of elements in short intervals of length roughly $x^{5/6}$ , thereby extending classical results on gaps between quadratic forms.

Original authors: Siddharth Iyer

Published 2026-05-19

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Siddharth Iyer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have two different types of "mathematical building blocks."

Block Type A (The Triangle Set): These are numbers you can build using the formula $x^2 + xy + y^2$ . Think of these as numbers that fit a specific triangular pattern.
Block Type B (The Square Set): These are numbers you can build by adding two perfect squares together (like $1^2 + 2^2 = 5$ ).

Now, imagine you have a magic ruler with a fixed length, let's call it $a$ . You want to find pairs of numbers where:

The first number is a Block Type A.
If you add your magic ruler length ( $a$ ) to it, the result is a Block Type B.

The paper is essentially a census (a headcount) of how many of these special pairs exist, and specifically, how far apart they can be.

The Big Question: How Big are the Gaps?

In the world of numbers, sometimes these special pairs appear very frequently, and sometimes there are long stretches where you find none. The author, Siddharth Iyer, is asking: "What is the maximum distance (gap) between two of these special pairs?"

If you walk along the number line looking for these pairs, how far do you have to walk before you are guaranteed to find the next one?

The Main Discovery

The paper proves that no matter how large a number you start with (let's call it $x$ ), you will never have to walk a "long" distance to find the next pair.

The Old Way: Previous mathematicians (like Estermann and Hooley) had figured out similar rules for other types of number patterns, but they were dealing with simpler cases.
The New Result: Iyer shows that for this specific mix of "Triangle" and "Square" numbers, the gap is surprisingly small.
- If you are at a huge number $x$ , the next valid pair is guaranteed to appear within a distance of roughly $x^{5/6}$ (which is a bit less than the square root of $x$ , but still a significant chunk) multiplied by a tiny bit of "noise" (a logarithmic factor).
- The Analogy: Imagine you are walking through a forest looking for rare flowers. The old rules said, "You might have to walk a mile to find the next one." Iyer's new rule says, "Actually, you will definitely find one within a few hundred steps, even if the forest is the size of a continent."

How Did They Do It? (The Toolkit)

To prove this, the author didn't just count numbers one by one (which would take forever). He used a sophisticated "mathematical telescope" built from three main tools:

The Arithmetic Telescope (Tolev's Theorem): This helps look at numbers that follow a specific rhythm (arithmetic progressions). It's like knowing that every 7th number has a special property.
The Multiplicative Lens (Blomer, Brüdern & Dietmann): This analyzes how numbers behave when you multiply them. It helps understand the "personality" of the numbers in the set.
The Character Filter: The author uses special mathematical filters (called characters) to separate the "good" numbers from the "bad" ones, ensuring the count is accurate.

The "Special Case" (The $a$ Factor)

The paper also looks at a specific scenario where the magic ruler length ( $a$ ) is a very special kind of number (one that can be written as $n^2 - 3m^2$ ).

If $a$ is special: The gap is even smaller (roughly the square root of $x$ ).
If $a$ is not special: The gap is slightly larger (roughly $x^{5/8}$ ), but still much smaller than the number $x$ itself.

Why Does This Matter?

The paper doesn't claim to cure diseases or build bridges. Its value is purely in the mathematical landscape.

It extends a classic result from 1932 by a mathematician named Estermann.
It confirms that these two different "shapes" of numbers (triangular and square-based) interact in a very predictable, dense way. They don't disappear into the void; they are always close by.

Summary in a Nutshell

Think of the number line as a long highway.

The Goal: Find spots where a "Triangle Car" is followed exactly $a$ miles later by a "Square Car."
The Finding: You will never have to drive more than a short, predictable distance to find the next pair of cars. They are packed relatively tightly together, even as the highway stretches to infinity.
The Method: The author used advanced number theory "gears" to prove that the gaps between these cars are mathematically bounded and small.

The paper is a victory for understanding the hidden order and density of numbers, proving that even in the vast, infinite ocean of integers, these specific patterns are never truly lost.

Technical Summary of "Gaps Between Quadratic Forms" by Siddharth Iyer

Problem Statement
This paper investigates the distribution of integers represented by specific quadratic forms within short intervals. Specifically, it addresses the problem of bounding the gaps between consecutive integers $n$ such that $n$ is represented by the quadratic form $x^2 + xy + y^2$ (denoted as the set $\triangle$ ) and $n+a$ is represented as a sum of two squares (denoted as the set $\mathbb{O}^2$ ), for a fixed non-zero integer $a$ . The set of interest is defined as $S(\triangle, \mathbb{O}^2, a) = \{n : n \in \triangle, n+a \in \mathbb{O}^2\}$ .

The work builds upon classical results concerning the correlation of representations of quadratic forms. It references Estermann's (1932) asymptotic formula for the sum of products of representation functions $r_2(n)r_2(n+a)$ , and subsequent improvements by Hooley, Chamizo, and Müller regarding the error terms and the density of such integers in intervals. While Lemma 1.2 establishes that gaps between elements of $S(\mathbb{O}^2, \mathbb{O}^2, a)$ are bounded by $O(x^{1/2})$ , the paper seeks to establish analogous, albeit weaker, bounds for the mixed case $S(\triangle, \mathbb{O}^2, a)$ .

Methodology
The author employs analytic number theory techniques, focusing on the asymptotic evaluation of sums involving multiplicative functions and character sums.

Representation Functions and Characters:
The paper utilizes the fact that the number of representations $R_2(n)$ for $x^2+xy+y^2$ and $r_2(n)$ for sums of two squares can be expressed via Dirichlet characters. Specifically, $R_2(n) = 6 \sum_{d|n} \chi_3(d)$ and $r_2(n) = 4 \sum_{d|n} \chi_4(d)$ , where $\chi_3$ and $\chi_4$ are non-trivial characters modulo 3 and 4, respectively.
Adaptation of Tolev's Theorem:
A central tool is a theorem by Tolev (2012) concerning sums of two squares in arithmetic progressions. The author adapts this to handle the correlation between $F_{\chi_3}(n)$ and $F_{\chi_4}(n+a)$ . However, a direct application of Müller's (1989) theorem on sums of multiplicative functions is insufficient because $\chi_3$ and $\chi_4$ cannot be simultaneously lifted to a primitive character modulo a common $k$ (specifically where $12|k$ ) without losing primitivity.
Analysis of Multiplicative Functions:
To overcome the limitation of non-primitive characters, the paper introduces a new approach involving the function $\eta_a(q)$ , which counts solutions to $\alpha^2 + \beta^2 \equiv a \pmod q$ . The core of the proof (Theorem 1.6) involves analyzing the sum:
$\sum_{\substack{n \le x \\ (n,b)=1}} F_{\psi}(n) F_{\chi_4}(n+a)$
where $\psi$ is a real non-trivial character modulo an even integer $b \ge 4$ . The author decomposes the sum using the properties of the convolution of $\psi$ with the constant function 1, denoted $F_\psi$ .
Behavior of $\eta_a(d)$ :
A significant portion of the technical work (Section 3.1) is dedicated to analyzing the behavior of the function $\lambda_a(d) = \eta_a(d)/d$ . The author establishes bounds and asymptotic properties for $\lambda_a(d)$ based on the prime factorization of $d$ and its relation to $a$ , utilizing lemmas regarding primes $p \equiv 1, 3 \pmod 4$ . This analysis is crucial for proving that the main term coefficient $\beta(\psi, a)$ is strictly positive.
Gap Analysis via Parametric Families:
For the gap result (Theorem 1.7), the paper constructs a parametric family of solutions to the equation $c^2 + 3d^2 + a = u^2 + v^2$ . By analyzing the gaps between values of a quartic polynomial derived from this family, the author derives bounds on the spacing of elements in $S(\triangle, \mathbb{O}^2, a)$ .

Key Contributions and Results

Theorem 1.6 (Asymptotic Formula): The paper proves an asymptotic formula for the sum of products of representation functions restricted to coprime conditions. It establishes that for a real non-trivial character $\psi$ modulo even $b \ge 4$ :
$\sum_{\substack{n \le x \\ (n,b)=1}} F_{\psi}(n) F_{\chi_4}(n+a) = \beta(\psi, a) \eta^*(\psi, a) x + O(x^{5/6} \log^{19} x)$
Crucially, the author proves that the leading coefficient $\beta(\psi, a)$ is strictly positive, ensuring the existence of such integers in the specified range.
Corollary 1.3 (Short Interval Census): Applying Theorem 1.6 with $\psi = \chi_6$ , the paper demonstrates that for any $\epsilon > 0$ , there exists a constant $H_a > 0$ such that for sufficiently large $x$ :
$\# S(\triangle, \mathbb{O}^2, a) \cap [x, x + H_a x^{5/6} \log^{19} x] \ge x^{5/6 - \epsilon}$
This confirms that the set $S(\triangle, \mathbb{O}^2, a)$ is dense enough to guarantee elements in intervals of length slightly larger than $x^{5/6}$ .
Theorem 1.7 (Gap Bound): The paper provides a bound on the largest gaps between elements of $S(\triangle, \mathbb{O}^2, a)$ . It shows that for $x \ge 1$ , there exists an element of the set in the interval $[x, x + D_a x^{\Upsilon(a)}]$ , where the exponent $\Upsilon(a)$ is:
- $1/2$ if $a$ can be represented as $n^2 - 3m^2$ for integers $n, m$ .
- $5/8$ otherwise.
  This result is described as a "weaker gap result" compared to the $x^{1/2}$ bound known for $S(\mathbb{O}^2, \mathbb{O}^2, a)$ .

Significance and Claims
The paper claims to extend the classical work of Estermann (1932) and the subsequent refinements by Müller (1989) to the context of mixed quadratic forms. By utilizing Tolev's theorem and conducting a detailed analysis of the multiplicative function $\eta_a(d)$ , the author successfully derives an asymptotic formula where previous methods (relying on primitive characters) failed.

The significance lies in establishing the existence of integers satisfying mixed quadratic representation conditions in short intervals, specifically proving that the gaps are not arbitrarily large. The author modestly notes that while a generalization of Theorem 1.7 is possible, it is not pursued here. Furthermore, the author acknowledges in the Acknowledgments that if future work utilizing spectral theory can better understand the interplay between distinct quadratic forms, the main result of this paper might become obsolete, indicating the current result is a stepping stone rather than a final resolution. The work also connects to the broader study of ideal norms in quadratic number fields, suggesting that similar results can be obtained for sets $S(\mathfrak{a}_K, \mathbb{O}^2, a)$ where $\mathfrak{a}_K$ represents norms of ideals in a quadratic field $K$ .