DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Sets defined by permitted binary digit positions

Definition

Let $N_{+} = {1, 2, \dots}$ and $S \subseteq N_{+}$ . Define the binary digit restriction set

$A_{S} = {\sum_{k = 1}^{\infty} a_{k} 2^{- k} : a_{k} \in {0, 1}, a_{k} = 0 if k \notin S}, a_{S} (n) = # (S \cap {1, \dots, n}) .$

Here $n \geq 0$ and $a_{S} (0) = 0$ . The series is the positive-start-index series of Series, partial sums, convergence and the sum, divergence, and the tail series, convergent since its tails are bounded by the geometric tails of For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges. In particular $A_{S} \subseteq [0, 1]$ and $0 \in A_{S}$ . The cardinality symbol uses Finite, countably infinite, countable, uncountable. Membership requires the existence of an allowed expansion; points with two binary expansions are retained if either expansion is allowed. No choice of a preferred expansion is part of this definition.

Depends on

Used by

Digit-position density determines Hausdorff dimension Proposition

Dependency tree · two levels

23 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 1.3.2 and Example 1.4.2 (standard reference, not scraped)