evidence boundary
supportsfrontiers / frontier
Erdős problems frontier
- 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
e1271/1271 · statement.attested · reviewer:will-blair · 2026-06-10 · null→null
Evidence atom
back to sources{"artifact_id":"va_9bc926d75e4e3881","artifact_packet_id":"cap_61973ee16b553d57","candidate_claim_id":"vc_001bc0c62be1eea6"}
- id
- vea_d16113258cd5c32e
- frontier
- Erdős problems frontier
- source
- vs_f48597abc0ad61ef
- finding
- vf_74448d988565c839
finding binding
boundopen_question
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.
source binding
source-boundcap_61973ee16b553d57 · vc_001bc0c62be1eea6
vs_f48597abc0ad61ef
review context
unverified1 events
1 reviewable changes and 0 evaluation records target this atom or its bound objects.
statement
{"artifact_id":"va_9bc926d75e4e3881","artifact_packet_id":"cap_61973ee16b553d57","candidate_claim_id":"vc_001bc0c62be1eea6"}
locator
span:0
extraction method
artifact_to_state_import
support relation
supports
condition refs
vcnd_206819241901b537
caveats
No caveats recorded.
Review, event, and evaluation records
2events
vev_40b1c008fd52365cfinding.assertedCandidate claim vc_001bc0c62be1eea6 imported from artifact packet cap_61973ee16b553d57
reviewer:erdos-db-trust · 2026-05-30
reviewable changes
vpr_50d4e38bb30e939dfinding.addCandidate claim vc_001bc0c62be1eea6 imported from artifact packet cap_61973ee16b553d57
applied · agent:erdos-spine-ingest · 2026-05-30
evaluations
No evaluation rows are attached.