Pith Number

pith:BUU3BFPG

pith:2026:BUU3BFPGYTXOM7OC3F22YHIEEE

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A note on arithmetic progressions with restricted differences

David Conlon, Huy Tuan Pham, Jacob Fox

When S is a large subset of the finite field containing zero, any subset of the n-dimensional vector space over the field that avoids three-term arithmetic progressions with differences in S to the n has size at most q to the power (1 minus

arxiv:2605.13628 v1 · 2026-05-13 · math.CO · math.NT

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Record completeness

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3 Author claim open · sign in to claim

4 Citations open

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Claims

C1strongest claim

if q is an odd prime power, there is ε_q>0 such that if S ⊆ F_q with 0 ∈ S and |S|>(q+1)/2 and A ⊆ F_q^n contains no three-term arithmetic progression whose common difference is in S^n, then |A| ≤ q^{(1-ε_q)n}.

C2weakest assumption

That Tao's slice rank method can be directly adapted to the setting of restricted differences without losing the polynomial rank bounds or requiring additional unstated conditions on S or q.

C3one line summary

Adapting the slice rank method yields that sets in F_q^n without 3-APs with differences in S^n have size at most q^{(1-ε_q)n} when |S|>(q+1)/2 and q is an odd prime power.

References

16 extracted · 16 resolved · 0 Pith anchors

[1] A. Bhangale, S. Khot, Y. P. Liu, and D. Minzer, On inverse theorems and combinatorial lines,FOCS 2025, 1672–1684 2025

[2] A. Bhangale, S. Khot, and D. Minzer, Effective bounds for restricted 3-arithmetic progressions inFn p, Discrete Anal.2024, Paper No. 16, 22 pp 2024

[3] J. Blasiak, T. Church, H. Cohn, J. A. Grochow, E. Naslund, W. F. Sawin, and C. Umans, On cap sets and the group-theoretic approach to matrix multiplication,Discrete Anal.2017, Paper No. 3, 27 pp 2017

[4] E. Croot, V. F. Lev, and P. P. Pach, Progression-free sets inZn 4 are exponentially small,Ann. of Math.185(2017), 331–337 2017

[5] J. S. Ellenberg and D. Gijswijt, On large subsets ofFn q with no three-term arithmetic progression, Ann. of Math.185(2017), 339–343 2017

Receipt and verification

First computed	2026-05-18T02:44:17.781766Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

0d29b095e6c4eee67dc2d975ac1d04212158527880204188eb74c98ab7610905

Aliases

arxiv: 2605.13628 · arxiv_version: 2605.13628v1 · doi: 10.48550/arxiv.2605.13628 · pith_short_12: BUU3BFPGYTXO · pith_short_16: BUU3BFPGYTXOM7OC · pith_short_8: BUU3BFPG

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/BUU3BFPGYTXOM7OC3F22YHIEEE \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 0d29b095e6c4eee67dc2d975ac1d04212158527880204188eb74c98ab7610905

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "fd89253f7df6642c7e44dba1933648e95f3df67b502b957ef8ee61bac9d75e99",
    "cross_cats_sorted": [
      "math.NT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-13T14:55:15Z",
    "title_canon_sha256": "5d55817c2d909f340d60333c7112be10af9d7ba0e09b2f25121503f04d33d7ba"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13628",
    "kind": "arxiv",
    "version": 1
  }
}