CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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An uncountable compact set can have dimension zero

Statement refuted

Assume the Axiom of Countable Choice. The implication “Hausdorff dimension zero forces countability” is false. For $S = {k^{2} : k \geq 1}$ , $A_{S}$ is compact and uncountable, yet $\dim_{H} A_{S} = 0$ and $H^{t} (A_{S}) = 0$ for every finite $t > 0$ .

Facts & Assumptions

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

[F1]

Under the standing Countable Choice hypothesis, the binary digit set is compact with dimension equal to the lower density of allowed positions; if the positions and their complement are infinite it is uncountable. Digit-position density determines Hausdorff dimension

[F2]

Every finite exponent strictly above Hausdorff dimension has zero Hausdorff measure. Hausdorff dimension is the unique critical exponent

Counterexample

1.1

Here $a_{S} (n) = ⌊ \sqrt{n} ⌋$ , so $a_{S} (n) / n \to 0$ . The digit theorem gives compactness and dimension zero. Both the square positions and the nonsquare positions are infinite, so its uncountability conclusion applies.

2.1

Every $t > 0$ lies strictly above this dimension. Hence $H^{t} (A_{S}) = 0$ for every such finite exponent. The conclusion concerns positive exponents only; at exponent zero the uncountable set is not null.

F2step 1.1∎

Depends on

Used by

Dimension zero forces countability False statement
Vanishing at all positive exponents forces countability False statement

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Bishop–Peres Example 1.4.2, square-position specialisation (standard reference, not scraped)