Claim from Rafikzeraoulia2025

$g (n) \leq n - 1 - c_{ε} n^{0.81668 - ε} .$

Stated by the author · rafikzeraoulia2025 · 2026-08-18

Problem link

ErdősProblems.com #653 · open the original problem page

Quoted from the source · Significance editor · 2026-08-16 · View source

Author involvement

Rafik Zeraoulia reviewed and approved the claim description, source version, hash, and reviewer map by correspondence on 2026-08-18.

Record note · rafikzeraoulia2025 · 2026-08-18

Summary · for readers

In one sentence: The manuscript gives a partial result, not a resolution: it reports an improved upper bound for the distance-count function in Erdős Problem #653.

Reader summary · Significance · 2026-08-16

Start here · for reviewers

Main deduction

Follow the improved upper bound from the restricted-center point-circle incidence argument to the missing distance-count values and exponent.

Theorem 1.1, Proposition 2.1, and Section 3 · The restricted-center incidence argument is the most delicate starting point. · Stated by the author · rafikzeraoulia2025 · 2026-08-18

Risk points

The delicate step is whether the selected low-R centers and their circle family satisfy every hypothesis of the restricted-center incidence theorem.
Theorem 2.2 and its use in Section 3 · Stated by the author · rafikzeraoulia2025 · 2026-08-18
The exponent improvement depends on the parameter bookkeeping surviving the passage from the incidence terms to beta(alpha) and theta.
Section 3, final optimization · Stated by the author · rafikzeraoulia2025 · 2026-08-18

Useful background

Basic point-circle incidence bounds and the pinned-distance definitions used in the problem statement.
Theorem 2.2 and the definitions preceding Proposition 2.1 · Stated by the author · rafikzeraoulia2025 · 2026-08-18

Needs checking

Verify the stated incidence hypotheses before spending time on the exponent algebra. — A failure here would affect the main deduction.
Theorem 2.2 application · Stated by the author · rafikzeraoulia2025 · 2026-08-18
Suggest another focused check

Review activity

Independent reruns: 0
Math assessments: 0
Written reviews: 0
Open checks: 3

What Significance checked: Significance fetched the proof-claim page and its linked Zenodo PDF, checked the Zenodo version metadata, and computed the PDF SHA-256 hash recorded here.

Still open: Significance has not independently checked the restricted-center incidence argument, the conversion to missing distance-count values, or the exponent calculation. Independent verification remains open.

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Copy and paste this summary.

The manuscript gives a partial result, not a resolution: it reports an
improved upper bound for the distance-count function in Erdős Problem #653.

Checked: Significance fetched the proof-claim page and its linked Zenodo
PDF, checked the Zenodo version metadata, and computed the PDF SHA-256 hash
recorded here.

Not checked: Significance has not independently checked the
restricted-center incidence argument, the conversion to missing
distance-count values, or the exponent calculation. Independent verification
remains open.

As of 2026-08-16T15:19:21Z (freshness: current)
Full record: https://hjyuh.github.io/significance/2026-rafikzeraoulia-erdos-653/
Significance records evidence. It does not judge the mathematics.

Scope: This is a partial result, not a resolution of Erdős Problem #653. Stated by the author · rafikzeraoulia2025 · 2026-08-18
Source: Centered-circle incidence bounds for planar distance-count spectra · zenodo-21858673-v1 · retrieved 2026-08-16T15:19:21Z
sha256 a87ca77b143f…

Evidence — 1 entry

Source inspection
ev-source-inspection

Public-source and version check only. This is not a mathematical review.

Significance fetched the queued proof-claim page, followed its direct Zenodo manuscript link, confirmed the public v1 metadata, and computed the PDF hash above. The page exposed no formal artifact and no substantive review comment about this claim.

Automated result · Significance editor · 2026-08-16 · View source

Plain-language explanation

This record is a review map, not a substitute for the manuscript: it identifies the exact claim, source version, and bounded open tasks so a reader can decide whether to inspect a particular passage. The paper's stated theorem is on the upper-bound side, while Problem #653 asks for a near-linear lower bound; those directions must not be blended.

Explanation · Significance

Plain-language explanation

This page records what the author claims and what Significance checked about the source file. It does not explain or judge the proof.

Explanation · Significance

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