Gaps between quadratic forms

Siddharth Iyer

arxiv: 2505.23428 · v1 · pith:SVXUC5YSnew · submitted 2025-05-29 · 🧮 math.NT

Gaps between quadratic forms

Siddharth Iyer This is my paper

Pith reviewed 2026-05-22 01:06 UTC · model grok-4.3

classification 🧮 math.NT

keywords quadratic formssums of two squaresshort intervalsarithmetic progressionsmultiplicative functionsEstermann theoremTolev theorem

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The pith

There exists H_a > 0 so that every interval of length H_a x^{5/6} log^{19}x contains at least x^{5/6-ε} integers n represented by x² + xy + y² with n + a a sum of two squares, for large x.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a short-interval lower bound for the size of the set S of integers n that are represented by the quadratic form x² + xy + y² while n + a is simultaneously a sum of two squares. For any fixed nonzero a it shows that a positive proportion of the expected density persists inside every interval of length proportional to x^{5/6} times a power of log x. The argument combines Tolev's theorem on the distribution of sums of two squares in arithmetic progressions with a detailed study of the multiplicative function that encodes the representation counts by the form x² + xy + y². A sympathetic reader cares because the result supplies a quantitative version of how densely these paired representations occur, extending the classical 1932 work of Estermann to a much finer scale.

Core claim

The paper proves that for every nonzero integer a there exists a constant H_a > 0 such that the cardinality of S(△, □₂, a) inside [x, x + H_a x^{5/6} log^{19} x] is at least x^{5/6 - ε} for all sufficiently large x. The proof proceeds by applying Tolev's 2012 theorem on sums of two squares lying in arithmetic progressions, after first analyzing the multiplicative function that appears in the work of Blomer, Brüdern and Dietmann (2009) to control the local densities arising from the form x² + xy + y². This supplies a short-interval strengthening of Estermann's 1932 result on the same set.

What carries the argument

Tolev's 2012 theorem on sums of two squares in arithmetic progressions, applied after the analysis of the multiplicative function studied by Blomer-Brüdern-Dietmann that encodes the singular series for the form x² + xy + y².

If this is right

The maximal gap between consecutive elements of S(△, □₂, a) is at most H_a x^{5/6} log^{19} x for large x.
The same short-interval counting method applies verbatim to any fixed shift a and yields a uniform positive lower bound inside those intervals.
The result gives a quantitative strengthening of the classical density theorems for the simultaneous representation by two distinct quadratic forms.
Analogous statements hold for the generalized set obtained by replacing the sum-of-two-squares condition with any other binary quadratic form to which Tolev-type theorems apply.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same technique would likely produce short-interval results for other pairs of quadratic forms once an analogous theorem on representations in arithmetic progressions is available.
Numerical checks counting the elements of S in sample intervals of length x^{5/6} for moderate x could test the sharpness of the exponent 5/6.
If stronger versions of Tolev's theorem become available with a smaller error term, the interval length in the main theorem could be reduced accordingly.

Load-bearing premise

That Tolev's theorem on sums of two squares in arithmetic progressions continues to hold without extra restrictions when the progressions are those determined by the representation counts of the form x² + xy + y².

What would settle it

An explicit sequence of arbitrarily large x together with an interval [x, x + c x^{5/6}] (for any fixed c) containing o(x^{5/6}) elements of S(△, □₂, a) would falsify the existence of such an H_a.

read the original abstract

Let $\triangle$ denote the integers represented by the quadratic form $x^2+xy+y^2$ and $\square_{2}$ denote the numbers represented as a sum of two squares. For a non-zero integer $a$, let $S(\triangle,\square_{2},a)$ be the set of integers $n$ such that $n \in \triangle$, and $n + a \in \square_{2}$. We conduct a census of $S(\triangle,\square_{2},a)$ in short intervals by showing that there exists a constant $H_{a} > 0$ with \begin{align*} \# S(\triangle,\square_{2},a)\cap [x,x+H_{a}\cdot x^{5/6}\cdot \log^{19}x] \geq x^{5/6-\varepsilon} \end{align*} 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, Br{\"u}dern \& Dietmann (2009). Our work extends a classical result of Estermann (1932) and builds upon work of M{\"u}ller (1989).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper improves the short-interval lower bound for n represented by x²+xy+y² with n+a a sum of two squares down to length x^{5/6} log^{19}x with x^{5/6-ε} elements, via Tolev and BBD multiplicative analysis.

read the letter

The main takeaway is that this paper gives a lower bound of x^{5/6-ε} many n in intervals of length H_a x^{5/6} log^{19}x where n is represented by x² + xy + y² and n + a is a sum of two squares. It reaches this scale by combining Tolev's 2012 theorem on sums of two squares in arithmetic progressions with the multiplicative function setup from Blomer-Brüdern-Dietmann 2009 to handle the representation conditions for the form with discriminant -3. This shortens the interval length compared to Müller 1989 while extending the classical Estermann 1932 result. The approach looks clean on the surface: they fix the right residue classes modulo small integers to satisfy the local conditions and then apply the prior theorem after analyzing the multiplicative function. The potential soft spot is uniformity. Tolev's result needs to deliver good error terms for the specific moduli and classes induced by the quadratic form, and the constants have to stay controlled at this interval length. The abstract asserts the analysis works, but the full proof would need to show the implied constants do not blow up or eat the main term. This is a focused technical piece for analytic number theorists interested in short-interval distribution of quadratic forms and sums of squares. A reader following refinements of Estermann-type problems would get something concrete from the parameter improvement. It deserves peer review to verify the error terms and the exact range where Tolev applies after the multiplicative analysis.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that for any nonzero integer a, the set S(△, □₂, a) of integers n represented by the quadratic form x² + xy + y² with n + a a sum of two squares satisfies a short-interval lower bound: there exists H_a > 0 such that the number of such n in [x, x + H_a x^{5/6} log^{19} x] is ≫ x^{5/6 - ε} for all sufficiently large x. The argument reduces the problem to an analysis of the multiplicative function encoding the representation condition (drawn from Blomer-Brüdern-Dietmann 2009) combined with an application of Tolev's 2012 theorem on the distribution of sums of two squares in arithmetic progressions, thereby extending Estermann's 1932 result and Müller's 1989 work.

Significance. If the uniformity and error-term details are verified, the result supplies a quantitative short-interval census for simultaneous representation by two distinct quadratic forms, which strengthens the classical literature on gaps and distribution in the image of quadratic forms. The explicit dependence on the fixed a and the use of established external theorems (with the multiplicative-function analysis as the novel linking step) constitute a clear technical contribution in analytic number theory.

major comments (1)

[Proof of the main theorem] The central reduction invokes Tolev (2012) on sums of two squares in arithmetic progressions after the Blomer-Brüdern-Dietmann (2009) multiplicative analysis. The moduli and residue classes are determined by the local conditions for the form x² + xy + y² (discriminant -3). The manuscript must explicitly confirm that these q lie inside the range where Tolev's error terms remain effective and uniform in a; otherwise the lower bound x^{5/6-ε} may fail to hold. (See the paragraph applying Tolev's theorem and the preceding analysis of the multiplicative function.)

minor comments (2)

[Abstract] The display equation in the abstract would benefit from a brief parenthetical reminder of the meaning of △ and □₂ for readers who skip the introduction.
[Introduction] Notation for the constant H_a should be introduced once in the introduction and used consistently; currently it appears only in the statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address the major comment below regarding the application of Tolev's theorem, providing clarifications and indicating where revisions will be made to strengthen the exposition.

read point-by-point responses

Referee: [Proof of the main theorem] The central reduction invokes Tolev (2012) on sums of two squares in arithmetic progressions after the Blomer-Brüdern-Dietmann (2009) multiplicative analysis. The moduli and residue classes are determined by the local conditions for the form x² + xy + y² (discriminant -3). The manuscript must explicitly confirm that these q lie inside the range where Tolev's error terms remain effective and uniform in a; otherwise the lower bound x^{5/6-ε} may fail to hold. (See the paragraph applying Tolev's theorem and the preceding analysis of the multiplicative function.)

Authors: We appreciate this observation. In the reduction, the moduli q arise from the local solubility conditions for the form of discriminant -3 (specifically, the 3-adic and other small prime conditions encoded in the multiplicative function from Blomer-Brüdern-Dietmann 2009) together with the fixed parameter a. Since a is a fixed nonzero integer, these moduli q are bounded by a constant depending only on a (in fact, q divides a fixed multiple of the discriminant 3). Tolev's 2012 theorem provides error terms that are effective and uniform for q up to x^θ with θ < 1/3 (or the precise range stated in that paper), which is amply satisfied by our fixed q as x → ∞. The lower bound x^{5/6-ε} therefore holds uniformly for the fixed a. To make this explicit, we will insert a short paragraph immediately after the application of Tolev's theorem verifying the size of q and citing the relevant range from Tolev (2012). revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent external theorems

full rationale

The paper's central result is obtained by applying Tolev's 2012 theorem on sums of two squares in arithmetic progressions together with an analysis of the multiplicative function from Blomer-Brüdern-Dietmann 2009. These are prior results by other authors whose statements are independent of the present work and do not depend on its conclusions. The abstract explicitly frames the argument as an extension of Estermann 1932 and Müller 1989 using these external inputs. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new free parameters or invented entities. It relies on standard facts about quadratic representation and on two external theorems whose statements are independent of the present work.

axioms (1)

standard math The integers represented by x² + xy + y² and by sums of two squares obey the classical arithmetic-progression and density properties used in Tolev's theorem.
Invoked when the authors apply Tolev 2012 to the relevant residue classes determined by the form x² + xy + y².

pith-pipeline@v0.9.0 · 5750 in / 1434 out tokens · 73484 ms · 2026-05-22T01:06:45.981275+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

R2(n) = 6 ∑_{d|n} χ3(d) ... ideal norm in Z[ω] for Q(√-3)

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

T. M. Apostol, ‘Introduction to analytic number theory’, Undergraduate Texts in Mathematics. New York-Heidelberg: Springer (1976). 12

work page 1976
[2]

Blomer, J

V. Blomer, J. Br¨ udern & R. Dietmann, ‘Sums of smooth squares’, Compos. Math. 145(6) (2009), 1401–1441. 9, 10

work page 2009
[3]

Chamizo, ‘Correlated sums of r(n)’, J

F. Chamizo, ‘Correlated sums of r(n)’, J. Math. Soc. Japan 51(1) (1999), 237–252. 2

work page 1999
[4]

Chamizo, ‘The additive problem for the number of representations as a sum of two squares’, Mediterr

F. Chamizo, ‘The additive problem for the number of representations as a sum of two squares’, Mediterr. J. Math. 19(1) (2022), 44. 2

work page 2022
[5]

Estermann, ‘An asymptotic formula in the theory of numbers’, Proc

T. Estermann, ‘An asymptotic formula in the theory of numbers’, Proc. London Math. Soc. 2(1) (1932), 280–292. 1, 2

work page 1932
[6]

Hooley, ‘On the intervals between numbers that are sum of two squares’, Acta Math

C. Hooley, ‘On the intervals between numbers that are sum of two squares’, Acta Math. 127 (1971), 279–297. 2

work page 1971
[7]

Hooley, ‘On the intervals between numbers that are sums of two squares

C. Hooley, ‘On the intervals between numbers that are sums of two squares. III.’, J. reine angew. Math. 267 (1974), 207–218. 2

work page 1974
[8]

Iwaniec, ‘Primes of the type φ (x, y)+ A where φ is a quadratic form’, Acta Arith

H. Iwaniec, ‘Primes of the type φ (x, y)+ A where φ is a quadratic form’, Acta Arith. 21 (1972), 203–234. 6

work page 1972
[9]

Luca & L

F. Luca & L. T´ oth, ‘The rth Moment of the Divisor Function: An Elemen- tary Approach’, J. Integer Seq. 20(7) (2017), 3. 9

work page 2017
[10]

M¨ uller, ‘On the asymptotic behaviour of the ideal counting function in quadratic number fields’, Monatsh

W. M¨ uller, ‘On the asymptotic behaviour of the ideal counting function in quadratic number fields’, Monatsh. Math. 108(4) (1989), 301–323. 3, 4, 6

work page 1989
[11]

D. I. Tolev, ‘On the Remainder Term in the Circle Problem in an Arithmetic Progression’, Proc. Steklov Inst. Math. 276 (2012), 261–274. 7

work page 2012
[12]

B. M. Wilson, ‘Proofs of some formulae enunciated by Ramanujan’, Proc. London Math. Soc. 21 (1922), 235-255. 9 School of Mathematics and Statistics, University of New South W ales, Sydney, NSW 2052, Australia Email address : siddharth.iyer@unsw.edu.au

work page 1922

[1] [1]

T. M. Apostol, ‘Introduction to analytic number theory’, Undergraduate Texts in Mathematics. New York-Heidelberg: Springer (1976). 12

work page 1976

[2] [2]

Blomer, J

V. Blomer, J. Br¨ udern & R. Dietmann, ‘Sums of smooth squares’, Compos. Math. 145(6) (2009), 1401–1441. 9, 10

work page 2009

[3] [3]

Chamizo, ‘Correlated sums of r(n)’, J

F. Chamizo, ‘Correlated sums of r(n)’, J. Math. Soc. Japan 51(1) (1999), 237–252. 2

work page 1999

[4] [4]

Chamizo, ‘The additive problem for the number of representations as a sum of two squares’, Mediterr

F. Chamizo, ‘The additive problem for the number of representations as a sum of two squares’, Mediterr. J. Math. 19(1) (2022), 44. 2

work page 2022

[5] [5]

Estermann, ‘An asymptotic formula in the theory of numbers’, Proc

T. Estermann, ‘An asymptotic formula in the theory of numbers’, Proc. London Math. Soc. 2(1) (1932), 280–292. 1, 2

work page 1932

[6] [6]

Hooley, ‘On the intervals between numbers that are sum of two squares’, Acta Math

C. Hooley, ‘On the intervals between numbers that are sum of two squares’, Acta Math. 127 (1971), 279–297. 2

work page 1971

[7] [7]

Hooley, ‘On the intervals between numbers that are sums of two squares

C. Hooley, ‘On the intervals between numbers that are sums of two squares. III.’, J. reine angew. Math. 267 (1974), 207–218. 2

work page 1974

[8] [8]

Iwaniec, ‘Primes of the type φ (x, y)+ A where φ is a quadratic form’, Acta Arith

H. Iwaniec, ‘Primes of the type φ (x, y)+ A where φ is a quadratic form’, Acta Arith. 21 (1972), 203–234. 6

work page 1972

[9] [9]

Luca & L

F. Luca & L. T´ oth, ‘The rth Moment of the Divisor Function: An Elemen- tary Approach’, J. Integer Seq. 20(7) (2017), 3. 9

work page 2017

[10] [10]

M¨ uller, ‘On the asymptotic behaviour of the ideal counting function in quadratic number fields’, Monatsh

W. M¨ uller, ‘On the asymptotic behaviour of the ideal counting function in quadratic number fields’, Monatsh. Math. 108(4) (1989), 301–323. 3, 4, 6

work page 1989

[11] [11]

D. I. Tolev, ‘On the Remainder Term in the Circle Problem in an Arithmetic Progression’, Proc. Steklov Inst. Math. 276 (2012), 261–274. 7

work page 2012

[12] [12]

B. M. Wilson, ‘Proofs of some formulae enunciated by Ramanujan’, Proc. London Math. Soc. 21 (1922), 235-255. 9 School of Mathematics and Statistics, University of New South W ales, Sydney, NSW 2052, Australia Email address : siddharth.iyer@unsw.edu.au

work page 1922