Pith Number

pith:4ZTJRY52

pith:2018:4ZTJRY52VV2ZH7BXWH3USUJ4GL

not attested not anchored not stored refs pending

A short note on the multiplicative energy of the spectrum of a set

Ilya D. Shkredov

arxiv:1805.10468 v1 · 2018-05-26 · math.NT · math.CO

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{4ZTJRY52VV2ZH7BXWH3USUJ4GL}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

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.

Receipt and verification

First computed	2026-05-18T00:14:52.420898Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e66698e3baad7593fc37b1f749513c32cac83149d63e887f9b3744c3fc2150c2

Aliases

arxiv: 1805.10468 · arxiv_version: 1805.10468v1 · doi: 10.48550/arxiv.1805.10468 · pith_short_12: 4ZTJRY52VV2Z · pith_short_16: 4ZTJRY52VV2ZH7BX · pith_short_8: 4ZTJRY52

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/4ZTJRY52VV2ZH7BXWH3USUJ4GL \
  | 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: e66698e3baad7593fc37b1f749513c32cac83149d63e887f9b3744c3fc2150c2

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "61c53ab8e1ed9bd588200e9c562c6f836a768a03e28f4beda707dca5fef9aec3",
    "cross_cats_sorted": [
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2018-05-26T12:12:37Z",
    "title_canon_sha256": "487dddafc2e1f99fd6ed2618f58fd5ce3c263bda2c4ea353ada241400696c97d"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1805.10468",
    "kind": "arxiv",
    "version": 1
  }
}