arxiv: 2604.18544 · v1 · submitted 2026-04-20 · 🧮 math.CA · math.CO· math.NT

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Near-optimal density theorems for large dilates of large point configurations

Adian Anibal Santos Sep\v{c}i\'c, Vjekoslav Kova\v{c}

Pith reviewed 2026-05-10 03:17 UTC · model grok-4.3

classification 🧮 math.CA math.COmath.NT

keywords density theoremspoint configurationsdilatessimilar copiesmeasurable setsequidistributionl^p norms

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The pith

Any measurable set in Euclidean space with density exceeding 1 minus O((log n)/n) contains all sufficiently large similar copies of any fixed n-point configuration.

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

The paper determines near-optimal density thresholds forcing Lebesgue measurable subsets of R^d to contain large scaled and rotated copies of arbitrary fixed n-point configurations. For any such configuration, density 1 minus a constant times (log n)/n suffices to guarantee similar copies at every sufficiently large scale. This bound matches the best known upper bounds except for the logarithmic factor and thereby resolves an open problem posed in earlier work. A parallel argument yields the sharper threshold 1 minus 1/n plus little-o terms for the same question in non-Euclidean l^p norms when p is not 2. The Euclidean proof combines equidistribution of polynomial sequences modulo 1 with probabilistic thinning, while the l^p case uses the geometry of those spaces.

Core claim

We prove a lower bound of the form 1-O((log n)/n) on the density threshold that forces a measurable set E in R^d to contain all sufficiently large similar copies of every n-point configuration. This matches the known upper bound up to the logarithmic factor. For embeddings into R^d equipped with the l^p norm, p in (1, infinity) excluding 2, the threshold is asymptotically 1-1/n+o(1/n).

What carries the argument

Equidistribution of polynomial sequences modulo 1 combined with probabilistic thinning, which selects large dilates inside sufficiently dense measurable sets; geometry of l^p spaces supplies the second estimate.

Load-bearing premise

The n-point configurations are fixed in advance, the set E is Lebesgue measurable, and the conclusion applies only to all sufficiently large dilates rather than every scale.

What would settle it

A Lebesgue measurable set E with upper density 1 - C(log n)/n for sufficiently large C that avoids similar copies of some fixed n-point configuration at all large scales would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.18544 by Adian Anibal Santos Sep\v{c}i\'c, Vjekoslav Kova\v{c}.

**Figure 1.** Figure 1: {y1, . . . , yn} is an isometric copy of {t1, . . . , tn} dilated by r. Cook, Magyar, and Pramanik [5], and later also in [6, 7], motivated by the fact that the results of those papers were not available in the Euclidean norm (that is, the ℓ 2 norm). Theorem 2. Fix integers d ⩾ 1 and p ⩾ 2. There exists a constant Cd,p ∈ (0, ∞) with the following property. For every sufficiently large integer n there exist… view at source ↗

**Figure 2.** Figure 2: The set E for p = 2. of length 1 − εn gives [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Configuration P in the proof of Theorem 3. for every j ⩾ 1. It remains to prove that E contains no ℓ p -isometric copy of rjP. Assume, to the contrary, that for some j a set Y ⊆ E is an ℓ p -isometric copy of rjP. By Lemma 7, there exist x, u ∈ R d with ∥u∥p = 1 such that points rjke1, k = −1, 0, 1, 2, . . . , n − 2d, are mapped to yk := x + rjku, k = −1, 0, 1, 2, . . . , n − 2d (4.1) via the aforementione… view at source ↗

read the original abstract

We study density thresholds that force a measurable set $E\subseteq\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration. We prove a lower bound of the form $1-O((\log n)/n)$, which matches the known upper bound up to the logarithmic factor, thus essentially resolving a problem posed by Falconer, Yavicoli, and the first author of the present paper. We also study the same problem for embeddings of $n$-point configurations into $\mathbb{R}^d$ equipped with the $\ell^p$ norm, obtaining an asymptotically sharp bound $1-1/n+o(1/n)$, as soon as $p\in(1,\infty)\setminus\{2\}$. In the proof of the former estimate we use equidistribution of polynomial sequences modulo $1$ combined with probabilistic thinning. The proof of the latter estimate relies on the geometry of the $\ell^p$ spaces for $p\neq2$.

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 nearly settles the open density threshold problem for large dilates of n-point configurations with a 1-O((log n)/n) lower bound via equidistribution and thinning.

read the letter

The main advance is a lower bound of 1 - O((log n)/n) on the density of a measurable set E in R^d that forces it to contain all sufficiently large similar copies of any fixed n-point configuration. This matches the known upper bound up to the log factor and directly addresses the question left open by Falconer, Yavicoli, and Kovač. The l^p-norm version gives the asymptotically sharp 1 - 1/n + o(1/n) for p not equal to 2, which is a clean extra result. The Euclidean proof combines equidistribution of polynomial sequences modulo 1 with probabilistic thinning to control uniformity across scales, while the l^p case uses the specific geometry of those norms. Both approaches look independent of the target bound and rely on standard tools. The assumptions (Lebesgue measurable E, configurations fixed in advance, large dilates only) are the usual ones and do not create circularity. The log factor is the remaining gap, but the paper states it plainly as near-optimal rather than claiming full optimality. Without the full derivations in front of me it is hard to inspect the precise error terms or thinning probabilities, yet nothing in the sketch suggests a load-bearing flaw. This is useful reading for people working on Falconer-type problems or density questions in geometric measure theory and combinatorial geometry. A serious editor should send it to peer review; the central claim is grounded enough to merit referee time even if the details need tightening.

Referee Report

0 major / 2 minor

Summary. The paper studies density thresholds for measurable sets E in R^d to contain all sufficiently large similar copies of every n-point configuration. It proves a lower bound of 1 - O((log n)/n), matching the known upper bound up to a log factor, essentially resolving a problem posed by Falconer, Yavicoli, and the first author. For l^p norms with p ≠ 2, it obtains an asymptotically sharp bound of 1 - 1/n + o(1/n). The proofs rely on equidistribution of polynomial sequences modulo 1 with probabilistic thinning for the Euclidean case and on the geometry of l^p spaces for the other case.

Significance. If the results hold, this work provides near-optimal density theorems for large dilates of point configurations, significantly advancing the field by essentially settling the problem in the Euclidean setting and providing sharp bounds in l^p. The combination of equidistribution and probabilistic methods is a strength, as is the geometric approach for non-Euclidean norms. The results are falsifiable and the bounds are explicit in their asymptotic form.

minor comments (2)

[Abstract] The phrase 'essentially resolving' in the abstract should be clarified by explicitly referencing the precise statement of the open problem from the cited work of Falconer, Yavicoli, and the first author.
[Introduction] The assumptions that E is Lebesgue measurable and that the n-point configurations are fixed in advance (with the result applying only to sufficiently large dilates) are standard but should be stated more prominently at the beginning of the introduction to frame the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for recommending minor revision. The referee's summary accurately captures the main results, including the near-optimal density threshold of 1 - O((log n)/n) in the Euclidean setting and the asymptotically sharp bound 1 - 1/n + o(1/n) for l^p norms with p ≠ 2.

Circularity Check

0 steps flagged

Minor self-citation in problem statement; derivation remains independent

full rationale

The paper establishes its central density lower bound of 1-O((log n)/n) via equidistribution of polynomial sequences modulo 1 combined with probabilistic thinning for the Euclidean case, and via geometric properties of ℓ^p norms for p≠2. These tools are external and independent of the target result. The only self-reference is noting that the open problem was posed by Falconer, Yavicoli, and the first author; this citation is limited to the problem statement and does not justify or reduce the proof steps themselves. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citation chains appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background results in measure theory and equidistribution; no free parameters or new postulated entities are introduced in the abstract.

axioms (2)

standard math Lebesgue measurability and density are well-defined for subsets of R^d
Implicit in the statement that E is measurable and has positive density.
standard math Equidistribution of polynomial sequences modulo 1 holds
Explicitly invoked in the proof sketch for the Euclidean case.

pith-pipeline@v0.9.0 · 5477 in / 1351 out tokens · 29464 ms · 2026-05-10T03:17:20.581259+00:00 · methodology

discussion (0)

Reference graph

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