DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

$(λ, t, m, d)$ -nets in base $b$

Definition

Let $b \geq 2$ , $d \geq 1$ , $m \geq 0$ , $0 \leq t \leq m$ , and $1 \leq λ < b$ be integers. An ordered point multiset of size $n = λ b^{m}$ in $[0, 1)^{d}$ is a $(λ, t, m, d)$ -net in base $b$ if every base- $b$ elementary interval $E = \prod_{j = 1}^{d} [a_{j} b^{- k_{j}}, (a_{j} + 1) b^{- k_{j}})$ with $\sum_{j = 1}^{d} k_{j} = m - t$ contains exactly $λ b^{t}$ entries, counting multiplicities. When $λ = 1$ this is the $(t, m, d)$ -net convention of Star discrepancy, digital nets, and digital sequences. In this item the parameters $b, t, λ$ are fixed when $m$ varies.

Depends on

Star discrepancy, digital nets, and digital sequences

Used by

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

Art B. Owen, Monte Carlo Variance of Scrambled Net Quadrature, §2 (standard reference, not scraped)
Art B. Owen, Monte Carlo Book: The Quasi-Monte Carlo Parts, Chapter 17 §17.5 (standard reference, not scraped)

(λ,t,m,d)(λ,t,m,d)-nets in base bb