Pith Number

pith:DSHKI22R

pith:2026:DSHKI22R3FAONEDCRYQXJBJOW4

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Extensions of the Furstenberg-S\'ark\"ozy theorem via the arithmetic level-$d$ inequality

Andrew Lott, Carlo Francisco E. Adajar, Chian Yeong Chuah, Krishnamohan Nandakumar, Mukul Rai Choudhuri, Nagendar Reddy Ponagandla, Rishika Agrawal, Steve Fan, Swaroop Hegde

For any intersective polynomial h the largest subset of {1,...,X} without nonzero h(n) differences has a quasipolynomial upper bound on size.

arxiv:2605.16216 v1 · 2026-05-15 · math.NT · math.CO

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Claims

C1strongest claim

We adapt their method to general intersective polynomials h∈Z[x] and obtain an analogous quasipolynomial upper bound for the largest subset of {1,2,…,X} whose difference set contains no nonzero element of the form h(n) with n∈Z.

C2weakest assumption

The arithmetic level-d inequality remains effective uniformly across all auxiliary polynomials arising in the iteration.

C3one line summary

Extends Furstenberg-Sárközy to general intersective polynomials h via uniform arithmetic level-d inequality, yielding the best known quasipolynomial density bound.

References

25 extracted · 25 resolved · 0 Pith anchors

[1] Arala,A maximal extension of the Bloom–Maynard bound for sets without square differences, Funct 2024

[2] A. Balog, J. Pelik´ an, J. Pintz, and E. Szemer´ edi,Difference sets withoutκth powers, Acta Math. Hungar. 65(1994), 165–187 1994

[3] T. F. Bloom and J. Maynard,A new upper bound for sets with no square differences, Compos. Math. 158(2022), 1777–1798 2022

[4] Davenport,Analytic methods for diophantine equations and diophantine inequalities, 2nd ed., Cam- bridge Mathematical Library, Cambridge University Press, 2005 2005

[5] R. Doyle, J. and A. Rice,Multivariate polynomial values in difference sets, Discrete Anal. (2021), no. 11, 46 pp 2021

Receipt and verification

First computed	2026-05-20T00:01:58.498795Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

1c8ea46b51d940e690628e2174852eb72163d1dcb6034b166b5282e6afdeaa59

Aliases

arxiv: 2605.16216 · arxiv_version: 2605.16216v1 · doi: 10.48550/arxiv.2605.16216 · pith_short_12: DSHKI22R3FAO · pith_short_16: DSHKI22R3FAONEDC · pith_short_8: DSHKI22R

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/DSHKI22R3FAONEDCRYQXJBJOW4 \
  | 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: 1c8ea46b51d940e690628e2174852eb72163d1dcb6034b166b5282e6afdeaa59

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b081ffbaa491fd57f996191f8ae2b4b6b2af9ae84383a348a8c8b110e9ac13b0",
    "cross_cats_sorted": [
      "math.CO"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-15T17:29:24Z",
    "title_canon_sha256": "ada7c1b45f35afc48b4a46069770661ce3eba1b174fae53ff7e215eb9e09204c"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16216",
    "kind": "arxiv",
    "version": 1
  }
}