Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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Critical Hausdorff measure is always finite and positive

Statement

Assume the Axiom of Countable Choice. The assertion “if dim⁡HA=s<∞, then 0<Hs(A)<∞” is false. Both the lower and upper strict inequalities can fail at dimension one.

Facts & Assumptions

Given: The objects, conventions, and hypotheses in the statement above.

[F1]

Under the standing Countable Choice hypothesis, the nonsquare-position digit set is compact of dimension one and has H1=0. A dimension-one set can have zero length

[F2]

Under the standing Countable Choice hypothesis, for every subset A⊆R, H1(A)=λ1∗(A). One-dimensional Hausdorff measure on the line is Lebesgue outer measure

[F3]

Under the standing Countable Choice hypothesis, dim⁡HR=1. Euclidean space and positive-volume sets have their Euclidean dimension

Refutation

1.1

The nonsquare-position digit set has dimension one but critical measure zero. Hence dimension alone does not force positive critical measure.

F1
2.1

The real line also has dimension one but critical measure infinity. Hence dimension alone does not force finite critical measure either. The two witnesses refute the two strict inequalities separately.

F2F3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources