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Besicovitch, Bisection, and the normality of $0.(1)(4)(9)(16)(25)\dots$

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Paul Pollack, Joseph Vandehey

Number Theory · Submitted 24 May 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We provide a new proof of the normality of the constant $0.(1)(4)(9)(16)(25)\dots$ formed by concatenating the squares, updating Besicovitch's methods.

how it is stated · asserted 1.00 read the record →

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arXiv ID1405.6266 (version 1)
Pith IDVQ4KANDM, Pith's permanent record of this paper and its review
CitationJoseph Vandehey, Paul Pollack. "Besicovitch, Bisection, and the normality of $0.(1)(4)(9)(16)(25)\dots$." arXiv:1405.6266 (2014). https://pith.science/paper/VQ4KANDM
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,977 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.25 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.92 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We revisit Besicovitch's 1935 paper in which he introduced several techniques that have become essential elements of modern combinatorial methods of normality proofs. Despite his paper's influence, the results he inspired are not strong enough to reprove his original result. We provide a new proof of the normality of the constant $0.(1)(4)(9)(16)(25)\dots$ formed by concatenating the squares, updating Besicovitch's methods.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We revisit Besicovitch's 1935 paper in which he introduced several techniques that have become essential elements of modern combinatorial methods of normality proofs. background 0.99 0.18 0.04 asserted 1.00
s1 Despite his paper's influence, the results he inspired are not strong enough to reprove his original result. gap 0.69 0.17 0.06 asserted 1.00
s2 We provide a new proof of the normality of the constant $0.(1)(4)(9)(16)(25)\dots$ formed by concatenating the squares, updating Besicovitch's methods. contribution 0.94 0.96 0.92 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We revisit Besicovitch's 1935 paper in which he introduced several techniques that have become essential elements of modern combinatorial methods of normality proofs. Despite his paper's influence, the results he inspired are not strong enough to reprove his original result. We provide a new proof of the normality of the constant $0.(1)(4)(9)(16)(25)\dots$ formed by concatenating the squares, updating Besicovitch's methods.

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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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