Pith Number

pith:F3S7D3S2

pith:2025:F3S7D3S2LGFTZR57JK3HJGXIDN

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Compression and complexity for sumset sizes in additive number theory

Melvyn B. Nathanson

For h-fold sumsets with large diameter, there is a compression algorithm to construct sets with the same sumset size but small diameter.

arxiv:2505.20998 v2 · 2025-05-27 · math.NT · math.CO

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

For sumsets hA with large diameter, there is a compression algorithm to construct sets A' with |hA'| = |hA| and small diameter.

C2weakest assumption

The claim that a compression procedure exists and works for arbitrary finite sets A subset of Z or Z^n whose h-fold sumset has large diameter, without additional restrictions on the structure of A or on the value of h and k (abstract, final paragraph).

C3one line summary

The paper introduces the sets R_Z(h,k) and R_{Z^n}(h,k) collecting all possible cardinalities of hA for |A|=k, studies their complexity, and supplies a diameter-compression algorithm that preserves |hA|.

References

12 extracted · 12 resolved · 1 Pith anchors

[1] Erd˝ os and E 1983

[2] J. Fox, N. Kravitz, and S. Zhang, Finer control on relative sizes of iterated susmets, preprint, 2025 2025

[3] G. A. Freiman,Foundations of a Structural Theory of Set Addition, American Mathematical Society, Providence, R.I., 1973 1973

[4] G. A. Freiman, What is the structure ofKifK+Kis small?, in:Number Theory, New York 1984–85, Springer-Verlag, New York, 1987, pages 109–134 1984

[5] Hegyv´ ari, On representation problems in the additive number theory, Acta Math 1996

Formal links

2 machine-checked theorem links

Cited by

1 paper in Pith

Additive sumset sizes with tetrahedral differences

Receipt and verification

First computed	2026-07-23T01:24:15.513961Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

2ee5f1ee5a598b3cc7bf4ab6749ae81b6d629d6cd8801fad23e66cd111e1bc98

Aliases

arxiv: 2505.20998 · arxiv_version: 2505.20998v2 · doi: 10.48550/arxiv.2505.20998 · pith_short_12: F3S7D3S2LGFT · pith_short_16: F3S7D3S2LGFTZR57 · pith_short_8: F3S7D3S2

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/F3S7D3S2LGFTZR57JK3HJGXIDN \
  | 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: 2ee5f1ee5a598b3cc7bf4ab6749ae81b6d629d6cd8801fad23e66cd111e1bc98

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "7962139064d6661bc146cac80da839e920c8199bcee5fccf93700600ffe2c3f9",
    "cross_cats_sorted": [
      "math.CO"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2025-05-27T10:32:11Z",
    "title_canon_sha256": "15cb4f447117a71d2bfa9707d90b126afc1cb903041258a40aa062823f6d019d"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2505.20998",
    "kind": "arxiv",
    "version": 2
  }
}