DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Equidistribution modulo one

Definition

A real sequence $(x_{n})_{n \geq 0}$ is equidistributed modulo one if, for every $0 \leq a \leq c \leq 1$ , the frequencies indexed by integers $n \geq 1$ satisfy

$\frac{1}{n} # {0 \leq k < n : {x_{k}} \in [a, c)} ⟶ c - a .$

Here ${x} = x - ⌊ x ⌋ \in [0, 1)$ , and the half-open convention makes the representative and both endpoints unambiguous. It includes the empty interval $a = c$ and the whole interval $[0, 1)$ .

Depends on

The circle, rotations and the doubling map

Used by

Initial terms of the square-root-two rotation Example
Weyl equidistribution fails for some rotation angles False statement
Weyl equidistribution for irrational rotations Theorem

Dependency tree · two levels

8 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

Charles Walkden, Ergodic Theory lecture notes (standard reference, not scraped)