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Paper metadata

Original submission
May 25, 2016
Latest revision
October 17, 2016 · version 3
Claimed MSC
arXiv categories
math.CO, math.NT

Abstract

Intrinsic Diophantine approximation on fractals, such as the Cantor ternary set, was undoubtedly motivated by questions asked by K. Mahler (1984). One of the main goals of this paper is to develop and utilize the theory of infinite de Bruijn sequences in order to answer closely related questions. In particular, we prove that the set of infinite de Bruijn sequences in k≥2k\geq 2 letters, thought of as a set of real numbers via a decimal expansion, has positive Hausdorff dimension. For a given kk, these sequences bear a strong connection to Diophantine approximation on certain fractals. In particular, the optimality of an intrinsic Dirichlet function on these fractals with respect to the height function defined by symbolic representations of rationals follows from these results.

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Audited against arXiv v3

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated September 26, 2026
01Statements2 reported findingsNo statement issues found

The quantitative Hausdorff-dimension bounds for uniformly de Bruijn expansions and their symbolic Diophantine consequences are correct.

Theorem 2.1No issue found

Uniformly de Bruijn expansions have positive but non-full dimension

Pages 3 and 5–11 · Theorem 2.1 and Section 4 · arXiv:1605.07953v3

The BEST-theorem count gives the strict upper bound log⁡(k!)klog⁡(k)δ\frac{\log(k!)}{k\log(k)}\delta. For k≥4k\geq4, the restricted Euler-tour construction gives the lower exponent log⁡((k−2)!)/(klog⁡(k))\log((k-2)!)/(k\log(k)); the separate binary and ternary extension constructions give the stated positive constants 1/491/49 and [8(9log⁡4(3)−1)]−1[8(9\log_4(3)-1)]^{-1}. The cylinder-measure estimates convert these combinatorial counts into the displayed Hausdorff-dimension bounds.

Full paper, version 3 ↗
Theorem 6.3 and Corollary 6.4No issue found

Uniformly de Bruijn points are badly symbolically approximable

Pages 15–16 · Theorem 6.3 and Corollary 6.4 · arXiv:1605.07953v3

A bounded gap in the de Bruijn orders forces any repeated preperiodic block of an approximating rational to be short relative to its period. With δ=log⁡(k)/log⁡(b)\delta=\log(k)/\log(b), the resulting inequality km−j−ℓ≤jk^{m-j-\ell}\leq j is exactly equivalent to b−m≥b−ℓ/(bjj1/δ)b^{-m}\geq b^{-\ell}/(b^j j^{1/\delta}). Strong separation converts the first differing digit into a uniform metric lower bound, proving bad symbolic approximation and the optimality consequence.

Full paper, version 3 ↗
02Proofs3 reported findingsNo proof issues found

The Euler-tour counts, extension constructions, mass-distribution estimates, and repeated-block argument are correct and complete after one harmless variable-name correction.

Sections 3–4No issue found in the reviewed proof

Eulerian extensions give the required branching estimates

Pages 4–11 · Theorem 3.4, Proposition 4.2, Proposition 4.5, and Corollaries 4.3 and 4.7 · arXiv:1605.07953v3

The paper's added Hamiltonian-cycle step, which repairs the earlier construction it cites, makes the residual de Bruijn graph regular. The BEST bijection then counts a controlled family of Euler tours. In the binary case the edge blocks EτE_\tau are disjoint, so changing a spanning tree inside the block indexed by each bit gives an injective family of at least 22n−22^{2^{n-2}} trees. These counts support the probability measures used in every lower-bound case, and the cylinder estimates satisfy the mass-distribution principle with the stated exponents.

Lemma 4.1Suspected typo

The concluding set name is mistyped

Page 5 · Hausdorff–Cantelli lemma · arXiv:1605.07953v3

The lemma defines S=lim sup⁡jUjS=\limsup_j U_j but concludes dim⁡H(U)≤s\dim_H(U)\leq s, where UU is undefined. Replace UU by SS. The proof and every later application already use the limsup set SS, so the correction is unique and has no mathematical effect.

Section 6No issue found in the reviewed proof

The symbolic-height estimate closes in both repetition regimes

Pages 15–16 · proof of Theorem 6.3 · arXiv:1605.07953v3

If the common prefix has length at most j+ℓj+\ell, the desired lower bound is immediate. Otherwise a de Bruijn order within the last ℓ\ell indices forbids the repeated period block from fitting in its de Bruijn prefix, giving kn≤m−n<j+ℓk^n\leq m-n<j+\ell. Since n≥m−j−ℓ+1n\geq m-j-\ell+1 and k≥2k\geq2, this yields km−j−ℓ≤jk^{m-j-\ell}\leq j and hence the asserted bound after using 1/δ=log⁡(b)/log⁡(k)1/\delta=\log(b)/\log(k).

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:1605.07953v3
Authors listed
Lior Fishman, Keith Merrill, David Simmons
Audit date
September 26, 2026
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