Pith Number

pith:QZ4VRXSI

pith:2026:QZ4VRXSIAA4STUOOSEBVVXNK4L

not attested not anchored not stored refs pending

Lebesgue measure of distance sets with regular pins and multi-scale Mizohata-Takeuchi-type estimates

Bochen Liu

If dim E >1, dim E + dim F >2 and F has equal Hausdorff and packing dimension, then some pinned distance set Delta_y(E) has positive Lebesgue measure.

arxiv:2603.15328 v3 · 2026-03-16 · math.CA · math.AP · math.CO

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\usepackage{pith}
\pithnumber{QZ4VRXSIAA4STUOOSEBVVXNK4L}

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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

✓ 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.

Claims

C1strongest claim

There exists y∈F such that Δ_y(E) has positive Lebesgue measure, given dim_H E>1, dim_H E + dim_H F>2, and F has equal Hausdorff and packing dimension.

C2weakest assumption

The assumption that F has equal Hausdorff and packing dimensions, which enables the multi-scale Good-Bad decomposition and Mizohata-Takeuchi estimates to control the distance set without extra losses.

C3one line summary

Under dim_H E >1, dim_H E + dim_H F >2 and F regular (equal Hausdorff and packing dimensions), there exists y in F such that the pinned distance set Δ_y(E) has positive Lebesgue measure.

Receipt and verification

First computed	2026-07-17T01:20:47.501936Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

867958de48003929d1ce91035addaae2c6087fdf775af57f7d8f9c31367db6d0

Aliases

arxiv: 2603.15328 · arxiv_version: 2603.15328v3 · doi: 10.48550/arxiv.2603.15328 · pith_short_12: QZ4VRXSIAA4S · pith_short_16: QZ4VRXSIAA4STUOO · pith_short_8: QZ4VRXSI

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/QZ4VRXSIAA4STUOOSEBVVXNK4L \
  | 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: 867958de48003929d1ce91035addaae2c6087fdf775af57f7d8f9c31367db6d0

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "f6fad89910c6a008ebe7ba6b57f52a48aeb97f85e639e189be40b77969ae8076",
    "cross_cats_sorted": [
      "math.AP",
      "math.CO"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
    "primary_cat": "math.CA",
    "submitted_at": "2026-03-16T14:20:56Z",
    "title_canon_sha256": "aaeac579701ef0901db5c42e7ddcc17f7745974c1258dc414029b64a2387e36c"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2603.15328",
    "kind": "arxiv",
    "version": 3
  }
}