Pith Number

pith:DLUZFPDE

pith:2026:DLUZFPDE653Y2D76CF3I3SYK52

not attested not anchored not stored refs resolved

Almost Affine Invariance Over Prime Fields: Green Problem 90

Jie Ma, Max Wenqiang Xu, Quanyu Tang

A density-1/2 set in the prime field F_p is almost affine invariant under all maps ax+b with |a| and |b| at most o(log p).

arxiv:2605.13454 v1 · 2026-05-13 · math.CO · math.DS · math.GR · math.NT

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Record completeness

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Claims

C1strongest claim

the threshold value of K such that A is almost affine invariant simultaneously under all ϕ(x) with |a|, |b|≤K and a≠0, is K=o(log p). This solves Ben Green's Open Problem 90.

C2weakest assumption

The o(p) error is uniform over all maps with |a|,|b|≤K; if the error term grows with K in a non-uniform way or depends on the specific density-1/2 assumption, the claimed threshold may fail to hold.

C3one line summary

The threshold K for almost affine invariance of density-1/2 sets in F_p is o(log p), solving Green's Problem 90.

References

8 extracted · 8 resolved · 0 Pith anchors

[1] S. Eberhard, B. Green, R. Mrazović,Translation and dilation invariance inZ/qZ, Unpublished notes

[2] Green,100 Open Problems, available athttps://people.maths.ox.ac.uk/greenbj/papers/ open-problems.pdf

[3] Some constructions in the inverse spectral theory of cyclic groups, 2003

[4] T. Hutchcroft and G. Pete,Kazhdan groups have cost 1, Invent. Math.221(2020), no. 3, 873–891.https: //doi.org/10.1007/s00222-020-00967-6 2020 · doi:10.1007/s00222-020-00967-6

[5] Freddie Manners,Is it known that(F × p ⋉F p, Fp)is not a relative expander family?, MathOverflow, URL (version: 2012-03-19):https://mathoverflow.net/q/91657 2012

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-18T02:44:41.843830Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

1ae992bc64f7778d0ffe11768dcb0aeea74cb823ef468ac06a25abefa6da3db8

Aliases

arxiv: 2605.13454 · arxiv_version: 2605.13454v1 · doi: 10.48550/arxiv.2605.13454 · pith_short_12: DLUZFPDE653Y · pith_short_16: DLUZFPDE653Y2D76 · pith_short_8: DLUZFPDE

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/DLUZFPDE653Y2D76CF3I3SYK52 \
  | 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: 1ae992bc64f7778d0ffe11768dcb0aeea74cb823ef468ac06a25abefa6da3db8

Canonical record JSON

{
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    "abstract_canon_sha256": "063e411d41543f57dc6713ce7e43a92c3acdbb5e60f1b716078042fdad185acd",
    "cross_cats_sorted": [
      "math.DS",
      "math.GR",
      "math.NT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-13T12:48:56Z",
    "title_canon_sha256": "9aeb54fad1ec50b8581c5f563d3fef4d086e519346b731c92e8d807488dbfdd5"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13454",
    "kind": "arxiv",
    "version": 1
  }
}