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arxiv: 2505.20851 · v2 · submitted 2025-05-27 · 🧮 math.CO · math.NT

Existence of a Sidon set for the distinct distance constant

Robin Riblet , Titien Schehr This is my paper

Pith reviewed 2026-05-19 13:32 UTC · model grok-4.3

classification 🧮 math.CO math.NT MSC 11B13
keywords Sidon setB_2[g]-setdistinct distance constantreciprocal sumcompactnessmaximizing setadditive combinatorics
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The pith

There exists a Sidon set whose sum of reciprocals equals the distinct distance constant.

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

The paper proves that Sidon sets can be chosen so the sum of the reciprocals of their elements reaches exactly the value of the distinct distance constant. This follows from showing that the collection of all Sidon sets is compact in a topology that makes the reciprocal-sum map continuous, hence a maximum is attained. A sympathetic reader cares because the distinct distance constant governs how few distinct distances a finite point set in the plane can produce, and a Sidon set achieving the matching sum supplies an explicit combinatorial object tied to that geometric bound. The same compactness also yields maximizers for other continuous functionals and improves the best numerical estimates known for the constant itself.

Core claim

By equipping the space of subsets of the positive integers with a topology in which the family of Sidon sets (and more generally B_2[g]-sets) is compact and the functional that sums the reciprocals of the elements is continuous, the authors obtain the existence of a Sidon set attaining the supremum of this sum; they identify that supremum with the distinct distance constant and derive sharper bounds on its numerical value.

What carries the argument

Compactness of the family of Sidon sets in a topology that makes the reciprocal-sum functional continuous, guaranteeing a maximizer exists.

If this is right

  • A Sidon set that maximizes the reciprocal sum exists and equals the distinct distance constant.
  • Numerical bounds on the distinct distance constant can be improved by analyzing properties of this maximizing set.
  • Maximizing sets exist for any continuous functional defined on Sidon sets or B_2[g]-sets.
  • The same compactness argument applies uniformly to all B_2[g]-sets for finite g.

Where Pith is reading between the lines

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

  • Finite Sidon sets with large reciprocal sums could be searched numerically to approximate the infinite maximizing set.
  • The link may let distance-geometry questions be rephrased as optimization problems over sparse additive bases.
  • Analogous existence statements might hold for other named constants arising from sum-distinct or sum-free sets.

Load-bearing premise

The collection of all Sidon sets is compact in a topology where the sum of reciprocals of elements changes continuously.

What would settle it

An explicit sequence of Sidon sets whose reciprocal sums converge to a number strictly smaller than the distinct distance constant, or a direct proof that the supremum cannot be attained inside the family.

read the original abstract

We highlight a certain compactness of Sidon sets and $B_2[g]$-sets and provide several applications. Notably, we prove the existence of such sets that maximize certain functions. In particular, we show the existence of a Sidon set whose reciprocal sum is equal to the distinct distance constant. We also improve the best known bounds for this constant.

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.

Referee Report

2 major / 2 minor

Summary. The paper establishes a compactness property for the collection of Sidon sets and B_2[g]-sets in the product topology on the power set of the natural numbers. It applies this compactness, together with continuity of the reciprocal-sum functional, to prove existence of a Sidon set that attains the supremum of the sum of reciprocals; this supremum is identified with the distinct-distance constant, and the paper also improves the best-known numerical bounds on that constant.

Significance. If the compactness and continuity arguments are correct, the work supplies a clean existence proof for an extremal Sidon set realizing the distinct-distance constant and yields a modest but concrete improvement on its numerical value. The approach recycles standard facts from topological dynamics on {0,1}^N (Tychonoff compactness, closedness of finite forbidden configurations, and uniform tail control of the harmonic series) and applies them to a concrete problem in additive combinatorics.

major comments (2)
  1. [§3] §3, paragraph following Definition 3.1: the argument that the Sidon property is closed relies on the fact that any forbidden 4-tuple equation involves only finitely many integers; this must be stated explicitly and the finite set of coordinates that determine membership in the limit must be identified.
  2. [Theorem 5.3] Theorem 5.3 and the paragraph preceding it: the identification of the maximal reciprocal sum with the distinct-distance constant is asserted but the precise relation between the two quantities (whether the constant is defined as this supremum or merely bounded by it) is not written as an equality with a reference to the original definition of the constant.
minor comments (2)
  1. [Introduction] The notation B_2[g] is used from the abstract onward but is defined only in §2; a one-sentence reminder in the introduction would help readers.
  2. [Table 1] Table 1 (numerical bounds) lists the new upper bound but does not indicate the computational method or the truncation level used to obtain it; a brief sentence or footnote would make the improvement reproducible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and will revise the manuscript accordingly to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [§3] §3, paragraph following Definition 3.1: the argument that the Sidon property is closed relies on the fact that any forbidden 4-tuple equation involves only finitely many integers; this must be stated explicitly and the finite set of coordinates that determine membership in the limit must be identified.

    Authors: We agree that the closedness argument benefits from greater explicitness regarding the finite support of each forbidden configuration. In the revised manuscript we will insert a sentence immediately after Definition 3.1 noting that any 4-tuple equation violating the Sidon property involves only finitely many natural numbers. We will identify this finite set of coordinates and explain that, in the product topology, membership of the limit set in the Sidon collection is completely determined by those coordinates, thereby confirming that the property passes to the limit. revision: yes

  2. Referee: [Theorem 5.3] Theorem 5.3 and the paragraph preceding it: the identification of the maximal reciprocal sum with the distinct-distance constant is asserted but the precise relation between the two quantities (whether the constant is defined as this supremum or merely bounded by it) is not written as an equality with a reference to the original definition of the constant.

    Authors: We thank the referee for this observation. While the manuscript already claims equality between the supremum of the reciprocal sums and the distinct-distance constant, we will make the relation fully explicit in the revision of Theorem 5.3 and the preceding paragraph. We will write the equality directly and add a reference to the original definition of the distinct-distance constant from the literature. revision: yes

Circularity Check

0 steps flagged

No significant circularity; compactness argument is self-contained

full rationale

The paper's central existence result for a Sidon set achieving the distinct distance constant follows from a standard compactness argument in the product topology on subsets of the naturals: the full power set is compact by Tychonoff, the Sidon (or B2[g]) property is closed because forbidden additive configurations involve only finitely many elements, and the reciprocal-sum functional is continuous because sets agreeing on an initial segment differ by at most a convergent tail. These facts are external to the target constant and do not reduce to any self-definition, fitted input, or self-citation chain. The derivation therefore remains independent and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on a compactness property of Sidon sets that is asserted rather than derived from more elementary facts in the abstract.

axioms (1)
  • domain assumption The family of Sidon sets is compact in a suitable topology making the reciprocal-sum functional continuous.
    Invoked to guarantee that a maximizing set exists.

pith-pipeline@v0.9.0 · 5570 in / 1133 out tokens · 36863 ms · 2026-05-19T13:32:27.616227+00:00 · methodology

discussion (0)

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Reference graph

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