arxiv: 2605.06621 · v1 · submitted 2026-05-07 · 🧮 math.CO · math.MG

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Point sets avoiding near-integer distances

Ritesh Goenka , Kenneth Moore

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Pith reviewed 2026-05-08 08:04 UTC · model grok-4.3

classification 🧮 math.CO math.MG

keywords point setsnear-integer distancesEuclidean ballcombinatorial geometrylifting lemmaSárközy constructionKonyagin boundsnowflaked embeddings

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The pith

In three dimensions, point sets avoiding near-integer distances can reach size X to the power 1 minus any epsilon for small enough delta, and achieve linear size in four dimensions.

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

The paper studies the largest possible number of points that can be placed inside a ball of radius X in d-dimensional space such that no two points have a distance within delta of any integer. Previous work in the plane gave constructions of size roughly the square root of X. The authors extend those constructions to three dimensions to obtain size nearly X for any epsilon when delta is small enough relative to epsilon. They introduce a lifting method that turns certain integer-distance sets into near-integer avoiding sets and use it to get linear size in four dimensions. They also adapt an earlier upper-bound technique to show that in any dimension the size is at most a constant times X to the power d over 2.

Core claim

Extending Sárközy's construction, we show that for every ε > 0, N_3(X, δ) = Ω_δ(X^{1-ε}) for δ sufficiently small in terms of ε. We also provide a lifting lemma from integer distance sets to sets avoiding near-integer distances via bilipschitz embeddings of snowflaked Euclidean spaces. This allows us to prove a linear lower bound N_4(X,δ) = Ω_δ(X) for all sufficiently small δ. Finally, adapting Konyagin's approach, we prove the upper bound N_d(X, δ) = O_{d, δ}(X^{d/2}) for all d ∈ ℕ.

What carries the argument

A lifting lemma that uses bilipschitz embeddings of snowflaked Euclidean spaces to transfer sets avoiding exact integer distances into sets avoiding near-integer distances.

If this is right

In three dimensions the largest such set has size at least X to the power 1 minus epsilon for any epsilon when delta is small enough.
In four dimensions the largest such set has size at least a positive constant times X when delta is sufficiently small.
In any dimension d the largest such set has size at most a constant depending on d and delta times X to the power d over 2.
These bounds partially answer questions of Erdős and Sárközy on the growth rate of such sets in dimensions greater than two.

Where Pith is reading between the lines

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

The lifting lemma could be applied to other problems of avoiding distances in specific sets rather than just the integers.
The remaining gap between the lower and upper exponents in dimensions three and higher indicates that tighter constructions or different upper-bound methods may still be possible.
Numerical checks of the lifting construction in low dimensions could reveal how small delta must be in practice for the linear bound to appear.

Load-bearing premise

The lower bounds require δ to be small enough depending on ε or the dimension, and the lifting step requires bilipschitz embeddings of snowflaked spaces that preserve the avoidance of near-integer distances.

What would settle it

An explicit 4-dimensional construction whose size grows linearly in X for some fixed positive δ independent of X, or a matching upper bound showing that N_4(X, δ) = o(X) for every δ > 0 as X tends to infinity.

Figures

Figures reproduced from arXiv: 2605.06621 by Kenneth Moore, Ritesh Goenka.

**Figure 1.** Figure 1: The construction with k = t = 3, projected onto the y-z plane, together with a section of the paraboloid z = 110(x 2 + y 2 ). The plot is shown at a limited resolution, so some nearby points appear visually clumped. Then for each such point p = (x (p) , y(p) , z(p) ), we have |x (p) | ⩽ k t+1 , |y (p) | ⩽ k t+1 , and |z (p) | ⩽ 8k 2t+4 , which implies |p| = q (x (p)) 2 + (y (p)) 2 + (z (p)) 2 ⩽ p 3(8k 2t+4… view at source ↗

read the original abstract

Let $d \in \mathbb{N}$, $\delta \in (0, 1/2)$, and $X > 0$. Denote by $N_d(X, \delta)$ the maximum number of points in a subset of the closed Euclidean ball of radius $X$ in $\mathbb{R}^d$ such that every pairwise distance is at least $\delta$ away from any integer. In the planar case, S\'ark\"ozy proved that for every $\varepsilon > 0$, $N_2(X, \delta) = \Omega_\delta(X^{1/2-\varepsilon})$ as $X \rightarrow \infty$ whenever $\delta$ is sufficiently small in terms of $\varepsilon$, while Konyagin proved the almost matching upper bound $N_2(X,\delta) = O_\delta(X^{1/2})$. We study this problem in higher dimensions, addressing a question of Erd\H{o}s and S\'ark\"ozy. Extending S\'ark\"ozy's construction, we show that for every $\varepsilon > 0$, $N_3(X, \delta) = \Omega_\delta(X^{1-\varepsilon})$ for $\delta$ sufficiently small in terms of $\varepsilon$. We also provide a lifting lemma from integer distance sets to sets avoiding near-integer distances via bilipschitz embeddings of snowflaked Euclidean spaces. This allows us to prove a linear lower bound $N_4(X,\delta) = \Omega_\delta(X)$ for all sufficiently small $\delta$. Finally, adapting Konyagin's approach, we prove the upper bound $N_d(X, \delta) = O_{d, \delta}(X^{d/2})$ for all $d \in \mathbb{N}$.

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

This paper gives the first almost-linear lower bound in 3D and a linear lower bound in 4D for near-integer distance avoiding sets, plus a uniform upper bound of order X^{d/2}, but the 4D claim depends on a lifting lemma whose details need checking.

read the letter

The core advance is the extension of Sárközy's planar construction to dimension 3, yielding N_3(X, δ) = Ω_δ(X^{1-ε}) for small δ, and then the use of a snowflaked bilipschitz lifting to reach a linear lower bound in dimension 4. The uniform upper bound N_d(X, δ) = O_{d,δ}(X^{d/2}) is also new and organizes the picture across dimensions. Both lower bounds address an explicit question of Erdős and Sárközy, and the upper bound adapts Konyagin's planar argument in a straightforward way. The lifting lemma itself is the genuinely new technical step; if it works as stated, it cleanly converts integer-distance sets into near-integer-avoiding ones in one higher dimension. That is useful and worth having on record. The dependence on δ being sufficiently small is explicit and acceptable for this type of result. The main soft spot is the lifting construction. The abstract indicates that α and the distortion must be chosen so the fractional parts stay away from integers, but it is not obvious from the summary whether the constants can be made independent of the cardinality of the base set. If the bilipschitz constant grows with the number of points or forces some distances into the forbidden δ-neighborhood for any fixed δ, the linear lower bound would collapse. The paper presumably addresses this in the proof, but that section is the one that requires the closest reading. The rest of the argument looks standard and free of circularity. This is for combinatorial geometers and people working on Erdős-type distance problems. A reader already familiar with the planar case will see the extensions quickly and can judge the lifting lemma on its own terms. The paper is coherent on its own terms and engages the literature directly, so it deserves a serious referee even if the 4D bound needs revision after checking the embedding details.

Referee Report

2 major / 2 minor

Summary. The paper defines N_d(X, δ) as the largest number of points inside a Euclidean ball of radius X in R^d such that every pairwise distance lies at least δ away from any integer. Extending Sárközy's planar construction, the authors prove N_3(X, δ) = Ω_δ(X^{1-ε}) for every ε > 0 whenever δ is sufficiently small in terms of ε. They introduce a lifting lemma that converts integer-distance sets into near-integer-avoiding sets via bilipschitz embeddings of snowflaked Euclidean metrics; this yields the linear lower bound N_4(X, δ) = Ω_δ(X) for all sufficiently small δ. Adapting Konyagin's method, they establish the general upper bound N_d(X, δ) = O_{d,δ}(X^{d/2}).

Significance. If the lifting lemma and the extended constructions hold, the work substantially advances the Erdős–Sárközy problem on point sets avoiding near-integer distances. The near-linear lower bound in dimension 3 and the linear lower bound in dimension 4 are notable improvements that approach the scale of the ambient space while respecting the distance restriction. The snowflake-embedding technique is a novel technical device with potential applications in metric embedding and combinatorial geometry. The dimension-dependent upper bound supplies a uniform benchmark against which the lower bounds can be compared.

major comments (2)

[Lifting lemma] Lifting lemma (central to the N_4 claim): the argument relies on composing a snowflake map d ↦ d^α (α < 1) with a bilipschitz Euclidean embedding. The manuscript must specify whether α and the distortion constant may be chosen independently of the cardinality of the input integer-distance set. If the distortion grows with set size, distances that were integers may be mapped into [n−δ, n+δ] for any fixed δ > 0, which would invalidate the claimed linear lower bound N_4(X, δ) = Ω_δ(X).
[3-dimensional lower bound] § on the 3-dimensional construction: the extension of Sárközy's method is asserted to produce N_3(X, δ) = Ω_δ(X^{1-ε}). The proof sketch should explicitly verify that the auxiliary parameters (e.g., the number of scales or the choice of δ relative to ε) remain uniform when the construction is lifted from the plane to R^3; otherwise the exponent 1−ε cannot be guaranteed for arbitrarily small δ.

minor comments (2)

[Abstract] The abstract states the 4-dimensional lower bound holds 'for all sufficiently small δ' while the 3-dimensional bound requires δ small in terms of ε. This distinction should be stated uniformly in the theorem statements and introduction.
[Notation and statements] Notation: the implied constants in the Ω_δ and O_{d,δ} statements depend on δ and d; making this dependence explicit in each theorem would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each of the major comments below.

read point-by-point responses

Referee: [Lifting lemma] Lifting lemma (central to the N_4 claim): the argument relies on composing a snowflake map d ↦ d^α (α < 1) with a bilipschitz Euclidean embedding. The manuscript must specify whether α and the distortion constant may be chosen independently of the cardinality of the input integer-distance set. If the distortion grows with set size, distances that were integers may be mapped into [n−δ, n+δ] for any fixed δ > 0, which would invalidate the claimed linear lower bound N_4(X, δ) = Ω_δ(X).

Authors: We appreciate the referee highlighting this important point regarding the independence of parameters in the lifting lemma. In our construction, the snowflake exponent α is selected depending solely on δ (for instance, α close to 1 but sufficiently less than 1 to ensure the image distances avoid integers by a margin proportional to δ), and this choice is independent of the cardinality of the integer-distance set. The bilipschitz embedding of the snowflaked metric into Euclidean space is achieved with a distortion constant that depends only on the dimension and α, hence independent of the set size, by appealing to known results on embedding snowflaked metrics (such as those from Assouad's theorem or similar embedding theorems for doubling metrics). For the specific integer-distance sets used in the proof (e.g., large subsets of integer lattices or other rigid structures), the embedding can be realized explicitly with uniform constants. We will revise the statement and proof of the lifting lemma to explicitly state these independence properties, thereby confirming that the mapped distances remain at least δ away from integers. revision: yes
Referee: [3-dimensional lower bound] § on the 3-dimensional construction: the extension of Sárközy's method is asserted to produce N_3(X, δ) = Ω_δ(X^{1-ε}). The proof sketch should explicitly verify that the auxiliary parameters (e.g., the number of scales or the choice of δ relative to ε) remain uniform when the construction is lifted from the plane to R^3; otherwise the exponent 1−ε cannot be guaranteed for arbitrarily small δ.

Authors: We agree that the current proof sketch in the section on the three-dimensional construction would benefit from greater explicitness regarding the uniformity of parameters. The extension from Sárközy's planar construction to R^3 involves iterating the planar construction across a third dimension while controlling the distances via a suitable discretization of scales. The number of scales is chosen as a function of ε only, and δ is taken sufficiently small depending on ε (but independent of X), ensuring that the error terms from the lifting do not accumulate to violate the distance condition. The exponent 1-ε arises from the planar exponent 1/2 - ε/2 or similar, combined with the linear factor in the third dimension. We will expand the proof to include a detailed verification of these parameter choices, making the uniformity explicit. revision: yes

Circularity Check

0 steps flagged

No circularity; bounds obtained from explicit constructions and independent lemmas

full rationale

The paper defines N_d(X, δ) externally as the maximum size of a point set in the ball of radius X with all pairwise distances at least δ from integers. Lower bounds for d=3 are obtained by extending Sárközy's explicit construction; the d=4 linear lower bound follows from a lifting lemma that composes snowflaking with a bilipschitz embedding of Euclidean space, mapping known integer-distance sets into the desired avoidance class. Upper bounds adapt Konyagin's combinatorial argument. None of these steps define N_d in terms of itself, rename a fitted parameter as a prediction, or reduce the central claim to a self-citation whose content is unverified. The lifting lemma invokes an external embedding theorem whose assumptions do not presuppose the target bound. The derivation chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rely on standard facts about Euclidean distance, the existence of bilipschitz embeddings of snowflaked metrics, and the validity of the earlier Sárközy and Konyagin arguments; no new free parameters or invented entities are introduced.

axioms (2)

standard math Euclidean distance satisfies the triangle inequality and is bilipschitz equivalent under snowflaking with exponent <1
Invoked in the lifting lemma to transfer integer-distance sets to near-integer-avoiding sets.
domain assumption Sárközy's planar construction and Konyagin's upper-bound argument extend verbatim to higher dimensions once δ is small enough
Central to the 3D lower bound and the general upper bound.

pith-pipeline@v0.9.0 · 5615 in / 1515 out tokens · 64897 ms · 2026-05-08T08:04:06.342377+00:00 · methodology

discussion (0)

Reference graph

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