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Erdős problems frontier

constellation seal · derived from vfr_37aec80d874a0239
id
vfr_37aec80d874a0239
license
CC-BY-4.0
findings
1,256
accepted core
6
contested
0
links
17
sources
1,234
evidence
1,256
avg conf
0.98

used by 0 · replayed by 2 producers

e1271/1271 · statement.attested · reviewer:will-blair · 2026-06-10 · null→null

Reviewable change

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verified — A frozen deterministic verifier re-checked the claim and passed.accepted

Erdős Problem #1097 remains OPEN. Statement: The main conjecture: for any finite set of integers $A$ with $|A| = n$, the number of distinct common differences in three-term arithmetic progressions is $O(n^{3/2})$. This conjecture was resolved negatively by showing that the problem is exactly equivalent to Bourgain's sums-differences question [Bo99], which was introduced as an arithmetic path towards the Kakeya conjecture. Under this equivalence: - The greatest achievable exponent for this problem is equal to the smallest constant $c$ achievable for Bourgain's sums-differences question: $$|A -_G B| \ll \max(|A|, |B|, |A +_G B|)^c$$ - The $O(n^{3/2})$ prediction is disproved because the lower bound has been shown to satisfy $c \ge 1.77898$ (due to Zheng and AlphaEvolve [GGTW25], improving on Lemm [Le15]), which is strictly greater than $3/2 = 1.5$. - The best known upper bound is $c \le 11/6 \approx 1.833$ (due to Katz and Tao [KaTa99]). - While the specific $O(n^{3/2})$ prediction is resolved negatively, the general question of determining the exact optimal exponent $c$ remains open. Topics: number theory, additive combinatorics. Erdős prize: no. Statement is machine-verified in Lean (formal-conjectures). OEIS: N/A.

id
vpr_50d4e38bb30e939d
frontier
Erdős problems frontier
kind
finding.add
created
2026-05-30
findings
+1
state
null → 6aead8da

accept gate

1 of 4 on record
—
signature
reviewer:erdos-db-trust · no key registered on this bundle
✓
chain
null → 6aead8da
—
witness
no verifier attachment on record for this target
—
grade
in state · unreviewed

timeline

  1. 2026-05-30proposeproposed · finding.addagent — machine actor, no signing keyagent:erdos-spine-ingestvpr_50d4e38bb30e939dCandidate claim vc_001bc0c62be1eea6 imported from artifact packet cap_61973ee16b553d57
  2. 2026-05-30acceptfinding.assertedreviewer:erdos-db-trustreviewer:erdos-db-trustnull→6aead8davev_40b1c008fd52365cCandidate claim vc_001bc0c62be1eea6 imported from artifact packet cap_61973ee16b553d57

proposed

reason

Candidate claim vc_001bc0c62be1eea6 imported from artifact packet cap_61973ee16b553d57

finding type

open_question

proposed confidence

0.99

confidence basis

agent-imported candidate claim; reviewer acceptance required

provenance

proposed by

agent — machine actor, no signing keyagent:erdos-spine-ingest

actor type

agent

created at

2026-05-30

target type

finding

affected

inspect finding →

Erdős Problem #1097 remains OPEN. Statement: The main conjecture: for any finite set of integers $A$ with $|A| = n$, the number of distinct common differences in three-term arithmetic progressions is $O(n^{3/2})$. This conjecture was resolved negatively by showing that the problem is exactly equivalent to Bourgain's sums-differences question [Bo99], which was introduced as an arithmetic path towards the Kakeya conjecture. Under this equivalence: - The greatest achievable exponent for this problem is equal to the smallest constant $c$ achievable for Bourgain's sums-differences question: $$|A -_G B| \ll \max(|A|, |B|, |A +_G B|)^c$$ - The $O(n^{3/2})$ prediction is disproved because the lower bound has been shown to satisfy $c \ge 1.77898$ (due to Zheng and AlphaEvolve [GGTW25], improving on Lemm [Le15]), which is strictly greater than $3/2 = 1.5$. - The best known upper bound is $c \le 11/6 \approx 1.833$ (due to Katz and Tao [KaTa99]). - While the specific $O(n^{3/2})$ prediction is resolved negatively, the general question of determining the exact optimal exponent $c$ remains open. Topics: number theory, additive combinatorics. Erdős prize: no. Statement is machine-verified in Lean (formal-conjectures). OEIS: N/A.

vf_74448d988565c839

Diff

Read-only frontier; diff not recomputed.

Review chain

  1. 01request

    Change request

    Erdős problems frontier receives a reviewable source, finding, caveat, replication, evaluation, or proof-affecting edit.

    open review →
  2. 02packet

    Diff packet

    The packet names affected record objects, evidence, rationale, reviewer-facing fields, and expected proof impact.

    open the campaign →
  3. 03checks

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  4. 04review

    Reviewer decision

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  5. 05accepted

    Accepted event

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finding.noted · reviewer:will-blair · 1 day

renders the record as of vev_d199cb2e · 1,338 events · hub

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