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arxiv: 1906.08437 · v1 · pith:ZNKXGCOJnew · submitted 2019-06-20 · 🧮 math.NT

Base phi representations and golden mean beta-expansions

Michel Dekking This is my paper

Pith reviewed 2026-05-25 19:52 UTC · model grok-4.3

classification 🧮 math.NT
keywords base-phi representationgolden meanbeta-expansionsBeatty sequencesdigit positionsnatural numbers
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The pith

Natural numbers with a 1 in the kth digit of their base-phi expansion are given by generalized Beatty sequences.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper gives exact descriptions of the natural numbers that have a 1 in any specified digit position within their base-phi representation. The representation writes every natural number uniquely as a sum of powers of the golden mean using only the digits 0 and 1, with the rule that no two 1s are adjacent. The descriptions take the form of generalized Beatty sequences and settle two earlier conjectures for the zeroth and first positions. A reader would see a concrete bridge between digit problems in a non-integer base and the classical construction of Beatty sequences.

Core claim

In the base phi representation any natural number is written uniquely as a sum of powers of the golden mean with digits 0 and 1, where the product of two consecutive digits is always 0. The sets of natural numbers for which the kth digit is 1 are generalized Beatty sequences, and the paper supplies precise expressions for these sets that prove two conjectures for k=0 and k=1.

What carries the argument

Generalized Beatty sequences that locate the natural numbers having a 1 in the kth position of their unique base-phi digit string.

If this is right

  • For every fixed k the numbers carrying a 1 at position k form a generalized Beatty sequence.
  • Closed-form expressions exist for these numbers in terms of the sequences.
  • The explicit formulas for k=0 and k=1 confirm the two stated conjectures.
  • The same sequence description applies uniformly to the golden-mean beta-expansion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Beatty-sequence description may allow direct computation of sums or averages over digits without enumerating expansions.
  • Similar position formulas could be sought for beta-expansions of other quadratic irrationals that admit a 0-1 digit set with a forbidden block.
  • The link suggests that certain Beatty sequences admit a direct combinatorial reading as digit-support sets in a fixed base.

Load-bearing premise

Every natural number possesses a unique base-phi representation that uses only digits 0 and 1 and never places two 1s next to each other.

What would settle it

A single natural number whose kth base-phi digit is 1 but which fails to belong to the generalized Beatty sequence claimed for that k.

read the original abstract

In the base phi representation any natural number is written uniquely as a sum powers of the golden mean with digits 0 and 1, where one requires that the product of two consecutive digits is always 0. In this paper we give precise expressions for the those natural numbers for which the $k$th digit is 1, proving two conjectures for $k=0,1$. The expressions are all in terms of generalized Beatty sequences.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript asserts that every natural number admits a unique base-phi representation as a sum of distinct powers of the golden mean with digits 0 and 1 subject to the no-adjacent-1s condition. It supplies explicit formulas, expressed via generalized Beatty sequences, for the set of natural numbers whose k-th digit equals 1, and proves the formulas for the cases k=0 and k=1, thereby establishing two conjectures.

Significance. If the derivations hold, the work furnishes concrete, closed-form descriptions of digit positions in the golden-mean beta-expansion. The explicit linkage to generalized Beatty sequences exploits the known partitioning properties of those sequences and supplies falsifiable, parameter-free characterizations that can be checked directly against the representation. This strengthens the interface between beta-expansions and combinatorial number theory.

minor comments (3)
  1. [Abstract] The uniqueness of the representation is invoked as a standard fact equivalent to Zeckendorf’s theorem; a brief citation to the relevant literature on Fibonacci numeration systems would clarify the foundation for readers outside the immediate subfield.
  2. The abstract states that the expressions are proved for k=0,1; the introduction or a dedicated section should restate the two conjectures verbatim before the proofs so that the reader can verify that the claimed statements match the conjectures exactly.
  3. Notation for the generalized Beatty sequences (e.g., the precise definition of the sequences B_{a,b} or their floor-function expressions) should be introduced once, with a displayed equation, before being used in the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on base-phi representations and the recommendation for minor revision. The referee summary correctly identifies the core results: unique representations with the no-adjacent-1s condition and explicit formulas via generalized Beatty sequences that prove the conjectures for k=0 and k=1. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states the uniqueness of the base-phi representation (no two consecutive 1s) as a given fact equivalent to the standard Zeckendorf theorem for Fibonacci numeration; this is an external mathematical result, not derived or fitted within the paper. The central claims consist of explicit expressions for the positions of 1-digits, expressed via generalized Beatty sequences, which are independent objects. No derivation step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the proofs for the k=0,1 cases rest on properties of Beatty sequences outside the paper's inputs. The work is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records the minimal background stated there. No free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Every natural number has a unique base-phi representation using digits 0 and 1 with the product of any two consecutive digits equal to zero.
    Stated directly in the abstract as the definition of the representation under study.

pith-pipeline@v0.9.0 · 5580 in / 1251 out tokens · 50320 ms · 2026-05-25T19:52:17.972439+00:00 · methodology

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Reference graph

Works this paper leans on

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