Close points on a modular hyperbola

Tsz Ho Chan

REVIEW 2 major objections 5 minor 9 references

Close points on a modular hyperbola

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves a first lower bound for the side of a square that must contain two points of a modular hyperbola: for infinitely many primes $p \equiv 1 \pmod{4}$ there is a residue $c$ modulo $p$ such that every square of side $\lfloor…

desk verdict Solid, well-executed extension in a niche area: new lower and upper bounds for close points on a modular hyperbola, with two fixable issues (Theorem 2's Z0 definition and the precise Graham–Ringrose statement). read the letter →

arxiv 2506.04087 v1 pith:V7CTMVBF submitted 2025-06-04 math.NT

Tsz Ho Chan This is my paper

classification math.NT MSC 11L4011T2411A15

keywords modularhyperbolacharactersumsLegendresymbolquadraticnonresiduessquarefreenumberssmoothalmostdensesetslowerbounds

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies solutions of $xy \equiv c \pmod{p}$, the points of a modular hyperbola, and asks how large a square must be before it is guaranteed to contain two of them. Its main result is a lower bound: for infinitely many primes $p \equiv 1 \pmod{4}$ there is a choice of $c$ for which every square of side $\lfloor \sqrt{0.1\log p}\rfloor$ still contains at most one point. This is the first lower bound of order $\sqrt{\log p}$ on the side length needed to force two points, and it comes from counting, with Weil's bound on character sums, values of $c$ that avoid small quadratic residues. The paper also proves upper bounds on the side length when the gaps between the two points must belong to special sets: multiplicatively closed sets, squarefree numbers, and almost dense sets such as the primes.

What carries the argument

The load-bearing identity is the Legendre-symbol criterion (3): two points $(x,y)$ and $(x+h,y+k)$ lie on $xy \equiv c \pmod{p}$ exactly when $\left(\frac{k}{p}\right)\!\left(\frac{k-4c\bar{h}}{p}\right)=1$ for some $1 \le h,k \le H$, where $\left(\frac{a}{p}\right)$ is $1$, $-1$, or $0$ according as $a$ is a square, a non-square, or $0$ modulo $p$. This converts the geometry of small squares into a character-sum counting problem. For Theorem 1, the counting is done with Weil's bound on products of Legendre symbols, combined with the Graham-Ringrose lower bound on the least quadratic nonresidue $n_p$, which guarantees that the interval of admissible $c$ is nonempty for infinitely many $p \equiv 1 \pmod{4}$. For Theorems 2 and 3, the same criterion is fed through Burgess-type estimates and Hölder's inequality with dense sets of $h$ and $k$; for Theorem 4, a double character-sum bound of Karatsuba type, in the sharpened form due to Chang, supplies the estimate.

What would settle it

Compute, for primes $p \equiv 1 \pmod{4}$ up to a large bound, the quantity $M(p) = \max_c m(p,c)$, where $m(p,c)$ is the least $H$ for which some square of side $H$ contains two points of $xy \equiv c \pmod{p}$. If $M(p) \le \lfloor \sqrt{0.1\log p}\rfloor$ for every tested prime, the infinite family promised by Theorem 1 would not appear in the data; a single prime with $M(p) > \lfloor \sqrt{0.1\log p}\rfloor$ would exhibit the claimed phenomenon and validate the counting mechanism.

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Extended reading notes

Core claim

The central claim is Theorem 1: there are infinitely many primes $p \equiv 1 \pmod{4}$ and integers $c$ with $\gcd(c,p)=1$ such that every square of side $\lfloor \sqrt{0.1\log p}\rfloor$ contains at most one point of the modular hyperbola $xy \equiv c \pmod{p}$. In other words, no universal guarantee of two points can be given with side length $o(\sqrt{\log p})$. The proof reduces the existence of two points $(x,y)$ and $(x+h,y+k)$ on the hyperbola to the condition that the Legendre symbols satisfy $\left(\frac{k}{p}\right)\!\left(\frac{k-4c\bar{h}}{p}\right)=1$ for some $1 \le h,k \le H$, then uses Weil's bound to count residues $c$ for which all small $h,k$ fail this test. The remaining theorems give upper bounds for restricted gaps: $H = C_{\delta,\epsilon}\,p^{1/4}\exp((\log p)^{1/2+\epsilon})$ for a multiplicatively closed set of positive density, the same shape for squarefree gaps (with one gap prime and the other a product of two primes as an option), and $H = C_{\epsilon,A}\,p^{11/34+\epsilon}$ for any almost dense set $A$, which covers prime gaps.

Load-bearing premise

The argument leans on a theorem of Graham and Ringrose, quoted without proof here, that infinitely many primes $p \equiv 1 \pmod{4}$ have a smallest non-square modulo $p$ at least of size $\log p\,\log\log\log p$; if that result were not available for this residue class, Theorem 1 would not follow from the given proof.

Editorial extensions

If this is right

No universal guarantee can force two points on $xy \equiv c \pmod{p}$ for every $c$ with a square of side $o(\sqrt{\log p})$; Theorem 1 puts a hard floor on the side length for an infinite family of primes.
For any multiplicatively closed set of positive density, two hyperbola points whose horizontal and vertical gaps lie in the set can be found in a square of side $p^{1/4}\exp((\log p)^{1/2+\epsilon})$, improving the earlier $p^{\epsilon}$ loss to a subexponential factor.
The same upper bound holds when the gaps are squarefree, and in that case one gap can even be taken prime while the other is a product of two distinct primes.
For any almost dense set, including the primes, two points with gaps in the set are guaranteed inside a square of side $p^{11/34+\epsilon}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extending the argument: the true threshold for the side length that forces two points for every $c$ lies somewhere between $\sqrt{\log p}$ and $p^{1/4}$, and the paper's lower bound suggests the answer may be a small power of $\log p$ rather than a power of $p$.
Because Theorem 1 inherits the Graham-Ringrose input, any improvement in unconditional lower bounds for least quadratic nonresidues in the residue class $1 \pmod{4}$ would immediately enlarge the constant $0.1$ and possibly extend the result to all primes.
The subexponential saving in Theorems 2 and 3 hints that the obstruction in the earlier $p^{1/4+\epsilon}$ result is concentrated on a thin set of bad gap pairs, so other naturally defined sets of gaps should admit similar savings through the same character-sum route.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Chan's paper has more in it than the title suggests: Theorem 1 is the first lower bound of order sqrt(log p) on the side length of a square that can contain two points on the modular hyperbola, for infinitely many primes, and Theorems 2-4 give genuinely new upper bounds when the two coordinates differ by numbers of special shape. The character-sum machinery is standard—Weil, Burgess, Karatsuba/Chang—but the reductions are clean and the applications are honest.

The main derivations check out. The reduction of two points to the product (hk/p)(hk-4c/p) is simple and correct, and the use of the Burgess moment method in Theorems 2 and 3 is fine; the subexponential factors cancel as advertised. Theorem 4's application of Chang's double character sum condition with k=3 works numerically. The self-citation [1] is used for context and the transformation is re-proven in Section 2, so no circularity.

The soft spots are real but fixable. In Theorem 2, the set Z0 is chosen as the larger of the quadratic residues or nonresidues in [1,T], without any intersection with M, despite the text claiming 'by pigeonhole principle and positive lower density of M.' As written, the proof does not guarantee that z0 z1 lies in M, so the contradiction to (6) is not licensed. The fix is to take Z0 as the larger of M∩[1,T]∩Qp or M∩[1,T]\Qp, which has size ≫δ T by the density assumption. This looks like an oversight rather than a deep flaw.

The bigger external dependency is Theorem 1. The character-sum estimate gives a c with the required nonresidue property once n_p > 0.1 log p, and Pólya–Vinogradov supplies n_p ≪ sqrt(p) log p, so the width of the interval works. But the infinite family relies entirely on Graham–Ringrose yielding infinitely many primes p ≡ 1 mod 4 with n_p ≫ log p log log log p. The paper cites [5] without quoting the statement. I believe their theorem is usually stated for all primes, and the construction can be adapted to p ≡ 1 mod 4, but this needs to be verified and the exact statement included. The stress-test note overstates it as 'unverified'; it's a citation hygiene issue unless the GR theorem truly doesn't cover the class.

Overall: the paper is a solid extension in a niche area, not a breakthrough. It deserves a serious referee, and I'd expect acceptance after a minor revision that fixes the Z0 definition and quotes the Graham–Ringrose theorem precisely.

Referee Report

2 major / 5 minor

Summary. The paper studies the distribution of points on the modular hyperbola xy ≡ c (mod p), specifically the side length H needed to force a square B_{X,Y}(H) to contain at least two points. Theorem 1 gives infinitely many primes p ≡ 1 (mod 4) for which some c satisfies that every square of side about sqrt(0.1 log p) contains at most one point, providing a lower bound of order sqrt(log p). Theorems 2 and 3 give upper bounds of the form p^{1/4} exp((log p)^{1/2+ε}) when the two distances h and k are required to lie in a multiplicatively closed set or in the squarefree numbers, and Theorem 4 gives an exponent 11/34 + ε for distances in an almost dense set. The proofs use character sums, Weil's bound, Burgess's moment method, and a double character sum bound of Karatsuba and Chang.

Significance. If the proofs are correct, Theorem 1 is a genuine improvement in the study of small squares on modular hyperbolas, providing the first lower bound of order sqrt(log p) for the side length needed to force two points for all c. The paper also gives a nice framework for distance-constrained variants, and the character-sum reductions in Section 2 are elegant and self-contained. The author is explicit about the standard character-sum ingredients, and the subexponential factors in Theorems 2 and 3 are designed to cancel correctly. However, the presentation of the proofs of Theorems 2 and 3 is sufficiently compressed that some steps are ambiguous, and one step in the reduction appears to involve the inverse of h rather than h itself.

major comments (2)

[Section 3, final paragraph] Theorem 1 depends on the assertion that Graham and Ringrose [5] supply infinitely many primes p ≡ 1 (mod 4) with n_p ≫ log p log log log p. The manuscript cites [5] without quoting the theorem, so the reader cannot verify that the required congruence class is covered. This is load-bearing because without infinitely many such primes in the class p ≡ 1 (mod 4), the infinite family in Theorem 1 would not follow. Please state the exact Graham–Ringrose theorem used and confirm that it applies to p ≡ 1 (mod 4); the standard form of that theorem does provide such primes, so this should be an explicit correction rather than a change of result.
[Section 4, around Eqs. (6)–(7)] The character sum S is written as Σ_{h∈A,z0∈Z0,z1∈Z1} χ(z0 − 4c h z1). The vanishing condition for this summand is h z0 z1 ≡ 4c (mod p), which corresponds to a pair with one distance equal to h z1. Since both h and z1 range up to H, the product h z1 can be as large as H^2, and H^2 = p^{1/2} exp((log p)^{1/2+ε}) is much larger than the theorem's claimed bound p^{1/4} exp((log p)^{1/2+ε}). As written, the argument therefore appears to prove the existence of two points whose one distance is bounded by H^2, not by H, and this would be a strictly weaker statement than Theorem 2. The same issue affects the proof of Theorem 3 in Section 5. Please clarify which variables are the final h and k, or use a character sum of the form χ(z0 z1 − 4c \bar h) whose vanishing condition is h z0 z1 ≡ 4c mod p, matching the claimed bound h ≤ H, k ≤ H T.

minor comments (5)

[Section 2, Eq. (3)] The second character condition should be (k/p)(k − 4c \bar h / p) = 1, with the multiplicative inverse of h, rather than k − 4ch; the first displayed form hk(hk − 4c) is correct, but the second form as written is missing the inverse. The same typo appears in Eq. (6) in Section 4.
[Introduction, Definitions] The phrase 'multiplicative closed set' should be 'multiplicatively closed set' in Definition 1 and in the statements of Theorems 2 and 3.
[Section 3, proof of Theorem 1] The sentence 'This together with (1) and (3)' appears to refer to the reduction (3), not the earlier theorem labelled (1); please correct the cross-reference.
[Theorem 1 statement] The notation 'gcd(c_p, p) = 1' is awkward; c is an integer, and the subscript in c_p appears only in the statement. Please use a single symbol consistently.
[Section 6, Theorem 4 proof] The word 'subet' should be 'subset'. Also, the 'almost dense' condition is stated only on dyadic intervals [X,2X] but is applied to the initial interval [1,p^α]; the deduction uses a standard dyadic summation and should be made explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's new results follow from standard external bounds and a self-contained character-sum reduction.

full rationale

The paper's derivation chain is self-contained after invoking standard external theorems. Theorem 1 reduces the two-point containment question to a character-sum condition in Section 2, and the transformation is derived directly from (x+h)(y+k) ≡ xy and completing the square, so the citation to the author's earlier work [1] is contextual rather than load-bearing. The lower bound for the character-sum average Σ is obtained by expanding products of Legendre-symbol indicators and applying Weil's bound (Theorem 5), with L = ⌊0.1 log p⌋. The only external input needed for the infinite family is the Graham–Ringrose lower bound n_p ≫ log p log log log p for infinitely many primes p ≡ 1 (mod 4), cited as [5]; this is an independent published external result, not a parameter fitted in this paper, and the paper does not claim to prove it. Theorems 2–4 use the same self-contained reduction together with Burgess's method and Karatsuba/Chang double character sum estimates, all external. There are no fitted inputs renamed as predictions, no uniqueness assertions imported from the authors' prior work, and no ansatz smuggled in via self-citation. Even if one worries about whether [5] supplies primes in the specific residue class p ≡ 1 (mod 4), that is a question about external support or correctness, not circularity. The paper meets the default expectation: no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard character sum techniques and on the Graham-Ringrose lower bound for least nonresidues. No new entities are introduced. No free parameters are fitted; the constants in the theorems are chosen to satisfy the inequalities in the proofs.

assumptions (5)

standard math Weil's bound for multiplicative character sums of non-square polynomials (Theorem 5)
Used in the proof of Theorem 1 and in the Burgess moment estimate in Theorems 2 and 3.
standard math Pólya-Vinogradov inequality for the least quadratic nonresidue n_p ≪ p^{1/2} log p
Invoked in the proof of Theorem 1 to ensure L < n_p is possible for the chosen L.
domain assumption Graham-Ringrose theorem: n_p ≫ log p log log log p for infinitely many primes p ≡ 1 (mod 4)
Cited in the proof of Theorem 1 to guarantee infinitely many primes where L = 0.1 log p is smaller than n_p. The paper does not prove this and relies on the cited result.
standard math Burgess's character sum moment bound for sums over intervals
Used in Theorems 2 and 3 to bound the high moments of the character sums over z0 and n.
standard math Double character sum bound of Karatsuba and Chang for a sum over an interval and an arbitrary set
Used in Theorem 4 to bound the sum over k in an interval and h in the almost dense set A.

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Cite this review

Pith. "Pith review of Close points on a modular hyperbola." pith.science (2026). https://pith.science/paper/V7CTMVBF

@misc{pith2026250604087,
  author       = {Pith},
  title        = {Pith review of: Close points on a modular hyperbola},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V7CTMVBF}},
  note         = {Machine review of arXiv:2506.04087}
}

abstract

In this paper, we continue the study of small squares containing at least two points on a modular hyperbola $x y \equiv c \pmod{p}$. We deduce a lower bound for its side length. We also investigate what happens if the ``distances" between two such points are special type of numbers like prime numbers, squarefree numbers or smooth numbers as well as more general multiplicatively closed sets or almost dense sets.

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Banks, M.Z

W.D. Banks, M.Z. Garaev and D.R. Heath-Brown, I.E. Shparlinski, Density of non-residues in Burgess-type intervals and applications, Bull. London Math. Soc. 40 (1) (2008), 88--96

work page 2008
[5]

Iwaniec and E

H. Iwaniec and E. Kowalski, Analytic number theory , AMS Colloquium Publications, Vol 53 (2004)

work page 2004
[2]

Chan, Shortest distance in modular hyperbola and least quadratic non-residue, Mathematika 62 (2016), 860--865

T.H. Chan, Shortest distance in modular hyperbola and least quadratic non-residue, Mathematika 62 (2016), 860--865

work page 2016
[3]

Chang, On a question of Davenport and Lewis and new character sum bounds in finite fields, Duke Math

M.C. Chang, On a question of Davenport and Lewis and new character sum bounds in finite fields, Duke Math. J. 145 (3) (2008), 409--442

work page 2008
[4]

Granville and K

A. Granville and K. Soundararajan, The spectrum of multiplicative functions, Ann. of Math. 153 (2) (2001), 407--470

work page 2001
[6]

Karatsuba, Distribution of values of Dirichlet characters on additive sequences, Soviet Math

A.A. Karatsuba, Distribution of values of Dirichlet characters on additive sequences, Soviet Math. Dokl. 44 (1) (1992), 145--148

work page 1992
[7]

Kerr, I.E

B. Kerr, I.E. Shparlinski and K.H. Yau, A refinement of the Burgess Bound for character sums, Michigan Math. J. 69 (2) (2020), 227--240

work page 2020
[8]

Graham and C.J

S.W. Graham and C.J. Ringrose, Lower Bounds for Least Quadratic Non-Residues. In: Berndt, B.C., Diamond, H.G., Halberstam, H., Hildebrand, A. (eds) Analytic Number Theory, Progress in Mathematics, vol 85 , Birkh\" a user Boston, MA, 1990, pp. 269--309

work page 1990

Show all 9 references

[9]

Shparlinski, Modular hyperbolas, Japan J

I.E. Shparlinski, Modular hyperbolas, Japan J. Math. 7 (2012), 235--294

2012

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Banks, M.Z

W.D. Banks, M.Z. Garaev and D.R. Heath-Brown, I.E. Shparlinski, Density of non-residues in Burgess-type intervals and applications, Bull. London Math. Soc. 40 (1) (2008), 88--96

work page 2008

[2] [5]

Iwaniec and E

H. Iwaniec and E. Kowalski, Analytic number theory , AMS Colloquium Publications, Vol 53 (2004)

work page 2004

[3] [2]

Chan, Shortest distance in modular hyperbola and least quadratic non-residue, Mathematika 62 (2016), 860--865

T.H. Chan, Shortest distance in modular hyperbola and least quadratic non-residue, Mathematika 62 (2016), 860--865

work page 2016

[4] [3]

Chang, On a question of Davenport and Lewis and new character sum bounds in finite fields, Duke Math

M.C. Chang, On a question of Davenport and Lewis and new character sum bounds in finite fields, Duke Math. J. 145 (3) (2008), 409--442

work page 2008

[5] [4]

Granville and K

A. Granville and K. Soundararajan, The spectrum of multiplicative functions, Ann. of Math. 153 (2) (2001), 407--470

work page 2001

[6] [6]

Karatsuba, Distribution of values of Dirichlet characters on additive sequences, Soviet Math

A.A. Karatsuba, Distribution of values of Dirichlet characters on additive sequences, Soviet Math. Dokl. 44 (1) (1992), 145--148

work page 1992

[7] [7]

Kerr, I.E

B. Kerr, I.E. Shparlinski and K.H. Yau, A refinement of the Burgess Bound for character sums, Michigan Math. J. 69 (2) (2020), 227--240

work page 2020

[8] [8]

Graham and C.J

S.W. Graham and C.J. Ringrose, Lower Bounds for Least Quadratic Non-Residues. In: Berndt, B.C., Diamond, H.G., Halberstam, H., Hildebrand, A. (eds) Analytic Number Theory, Progress in Mathematics, vol 85 , Birkh\" a user Boston, MA, 1990, pp. 269--309

work page 1990