Spectrum of the refined Diophantine exponent

Abstract: The refined Diophantine exponent, recently introduced by the author, is a quantity that measures the periodicity of an infinite word. In this article, we study this exponent from combinatorial and topological viewpoints. First, we show that, over a ternary alphabet, the spectrum of the refined Diophantine exponent is $[1,\infty]$. Second, we show that this exponent has topological properties similar to those of the set of Liouville numbers. Finally, we provide concrete examples with the Champernowne, Rudin--Shapiro, and Thue--Morse words, words coming from coding a rotation by intervals, and bracket words.

• The paper determines the spectrum of the refined Diophantine exponent $\mathbf{Rdio}$ over ternary alphabets, establishing it as $[1, \infty]$.
• It also shows the refined exponent can tolerate bounded mismatches, formalized via $(\epsilon, \delta)$-closeness, making it more permissive for automatic sequences.
• The paper proves measure-theoretic and topological properties of $\mathbf{Rdio}$, analogously to Liouville numbers, and computes specific values for Champernowne, Thue–Morse, and Rudin–Shapiro words, among others.

Overview

This paper, by Quang-Khai Nguyen (2608.20191), studies the refined Diophantine exponent $\mathbf{Rdio}(\mathbf{a})$ of an infinite word $\mathbf{a}$, a quantity introduced by the author in prior work as a refinement of the Diophantine exponent of Adamczewski and Bugeaud (Nguyen, 28 May 2026). Whereas the classical exponent detects repetitions $U_nV_n^w$ occurring as prefixes with $|V_n|\to\infty$, the refined version permits an arbitrarily small proportion $\epsilon$ of mismatches between the two compared blocks, formalized via $(\epsilon,\delta)$-closeness: the mismatching positions must be coverable by at most $\delta$ intervals whose total length is at most $\epsilon L$. The quantity measures how far a word is from being eventually periodic while tolerating bounded "noise," and it drives a transcendence criterion: if $\mathbf{a}$ is written over algebraic numbers and $\beta$ is algebraic with $\mathbf{a}$0 and $\mathbf{a}$1, then $\mathbf{a}$2 is either in $\mathbf{a}$3 or transcendental, by Schmidt's Subspace Theorem.

The paper makes three contributions: it determines the spectrum of $\mathbf{a}$4 over ternary alphabets; it establishes measure-theoretic and topological properties analogous to those of Liouville numbers; and it computes or bounds the exponent for concrete families — Champernowne, Rudin–Shapiro, Thue–Morse words, codings of irrational rotations by intervals, and bracket words.

Full spectrum over ternary alphabets

The first main result states that over a ternary alphabet the spectrum of $\mathbf{a}$5 is exactly $\mathbf{a}$6. The proof proceeds in two steps. First, the author establishes an existence result via the probabilistic method: there exists an infinite word $\mathbf{a}$7 over $\mathbf{a}$8 such that for all $\mathbf{a}$9, $U_nV_n^w$0, $U_nV_n^w$1,

$U_nV_n^w$2

The argument applies the Azuma–Hoeffding inequality to the martingale $U_nV_n^w$3 under the natural filtration, then takes a union bound over all triples $U_nV_n^w$4 with $U_nV_n^w$5; the tail probabilities sum to less than 1. This yields a linear lower bound on the mismatch count $U_nV_n^w$6 whenever $U_nV_n^w$7 — a substantially stronger effective pseudorandomness condition than what is available in the Mauduit–Sárközy literature, which is essential for the construction.

Second, given any $U_nV_n^w$8, the author sets $U_nV_n^w$9 and interleaves blocks of zeros of lengths $|V_n|\to\infty$0 with consecutive length-$|V_n|\to\infty$1 blocks of $|V_n|\to\infty$2, producing a word $|V_n|\to\infty$3 over $|V_n|\to\infty$4. The lacunary structure forces $|V_n|\to\infty$5 exactly, and the pseudorandomness of $|V_n|\to\infty$6 rules out any near-repetition achieving a larger ratio: any candidate block spanning a whole block of $|V_n|\to\infty$7 would incur $|V_n|\to\infty$8 mismatches against a permitted excess of only $|V_n|\to\infty$9 in length, contradicting $\epsilon$0-closeness. Hence $\epsilon$1, and since $\epsilon$2 is arbitrary together with known examples attaining $\epsilon$3 and $\epsilon$4, the spectrum is all of $\epsilon$5.

A corollary shows that over a quaternary alphabet one can realize $\epsilon$6 strictly, demonstrating concretely that the refined exponent is genuinely more permissive than the classical one. This is precisely the mechanism by which the Subspace-Theorem-based criterion of (Nguyen, 28 May 2026) applies to words beyond the reach of earlier criteria for Sturmian, morphic, and related words (Luca et al., 2023, Kebis et al., 2024).

Measure and category

Two results mirror the classical dichotomy for Liouville numbers (measure zero but residual). Under the uniform Bernoulli product measure on $\epsilon$7 for finite $\epsilon$8, the author proves

$\epsilon$9

via Borel–Cantelli: the probability that a fixed shift $(\epsilon,\delta)$0 admits a repetition factor exceeding $(\epsilon,\delta)$1 is summable in $(\epsilon,\delta)$2. Since $(\epsilon,\delta)$3 iff $(\epsilon,\delta)$4, this says almost every word has minimal refined exponent — so the Subspace-Theorem strategy cannot apply to random words. In the Cantor topology, using shift-invariance of $(\epsilon,\delta)$5 (proved here), the set of words with any prescribed value $(\epsilon,\delta)$6 is dense in $(\epsilon,\delta)$7 for $(\epsilon,\delta)$8: one prepends an arbitrary cylinder-defining prefix to a witness word from the spectrum theorem. Both results also hold for the classical $(\epsilon,\delta)$9. The density theorem inherits the hypothesis $\delta$0 from the spectrum theorem, which is currently established only for ternary and larger alphabets.

Explicit words: Champernowne, Thue–Morse, Rudin–Shapiro

For the Champernowne word, the paper proves $\delta$1. The proof does not use normality; instead, it exploits the fact that certain marker blocks $\delta$2 occur uniquely among integers of $\delta$3 and $\delta$4 digits, which caps the length of any prefix power $\delta$5 by roughly $\delta$6 against a prefix length of order $\delta$7.

For the Thue–Morse word $\delta$8 ($\delta$9), the paper establishes $\epsilon L$0. The key estimate, proved by induction on $\epsilon L$1 using the recurrences $\epsilon L$2, $\epsilon L$3, is

$\epsilon L$4

which extends to $\epsilon L$5 for all $\epsilon L$6 by locating an aligned dyadic sub-block inside $\epsilon L$7. Combining this with the definition of Condition $\epsilon L$8 gives $\epsilon L$9, hence $\mathbf{a}$0. Notably, the same computation shows $\mathbf{a}$1, so the Thue–Morse word fails the strong pseudorandomness condition of the spectrum construction — its finiteness of $\mathbf{a}$2 rests on the weaker linear bound $\mathbf{a}$3. The Rudin–Shapiro word is handled analogously: induction gives $\mathbf{a}$4 for the two correlation sums, yielding the same bound $\mathbf{a}$5, improving on the trivial consequence $\mathbf{a}$6 inherited from its critical exponent. These are upper bounds only; exact values remain open.

Codings of rotations by intervals

Let $\mathbf{a}$7 where $\mathbf{a}$8 is non-constant piecewise-constant with $\mathbf{a}$9 partition intervals and $\beta$0 irrational. The paper proves $\beta$1 unconditionally. Using continued-fraction convergents $\beta$2, the law of best approximation implies that within any window of length $\beta$3, at most $\beta$4 indices can be mismatches relative to shift $\beta$5 (each boundary interval $\beta$6 of width $\beta$7 contains at most one orbit point). Covering a block of length $\beta$8 by windows of length $\beta$9 bounds total mismatches by $\mathbf{a}$00, independent of $\mathbf{a}$01, so Condition $\mathbf{a}$02 holds for every $\mathbf{a}$03.

The classical exponent exhibits a dichotomy: if $\mathbf{a}$04 is badly approximable then $\mathbf{a}$05 (finiteness follows because the index of $\mathbf{a}$06 is finite, shown via discrepancy estimates split into cases according to whether $\mathbf{a}$07 exceeds the minimum partition gap $\mathbf{a}$08); if $\mathbf{a}$09 is well approximable then $\mathbf{a}$10, via a telescoping argument showing $\mathbf{a}$11 along convergent denominators. The motivation comes from degree sequences of monomial surface self-maps on projective toric surfaces [(Nguyen, 28 May 2026) context]: reductions modulo primes of such degree sequences reduce to evaluating piecewise-linear functions along orbits of Gaussian integers, a dynamics not captured by bracket words.

Bracket words

Bracket words are evaluations $\mathbf{a}$12 of finitely-valued generalized polynomials. For polynomial $\mathbf{a}$13 of degree $\mathbf{a}$14 with $\mathbf{a}$15, the paper proves:

• Finiteness: if the leading coefficient $\mathbf{a}$16 is badly approximable, then $\mathbf{a}$17. The proof controls the two-dimensional discrepancy of the sequence $\mathbf{a}$18 via Erdős–Turán–Koksma. Frequencies with $\mathbf{a}$19 are handled by Weyl's equidistribution theorem; frequencies with $\mathbf{a}$20 lead to exponential sums of degree $\mathbf{a}$21 polynomials with leading coefficient proportional to $\mathbf{a}$22, controlled by Weyl's quantitative bound combined with Dirichlet approximation and bad approximability of $\mathbf{a}$23. Optimizing the Dirichlet parameter $\mathbf{a}$24 yields $\mathbf{a}$25, uniformly in $\mathbf{a}$26, which contradicts the forced positive density $\mathbf{a}$27 of mismatches when $\mathbf{a}$28 is large.
• Infinite Dio under a strong Diophantine condition: if there exist $\mathbf{a}$29 and $\mathbf{a}$30 with $\mathbf{a}$31 and $\mathbf{a}$32, then $\mathbf{a}$33. Here $\mathbf{a}$34 splits into a rational part (periodic mod 1 with period $\mathbf{a}$35) and a remainder bounded by $\mathbf{a}$36 on the relevant range, so mismatches force real roots of a degree-$\mathbf{a}$37 polynomial, bounding their count by $\mathbf{a}$38; telescoping again gives $\mathbf{a}$39.
• Monomial case: for $\mathbf{a}$40 with rational partition boundaries, the weaker condition $\mathbf{a}$41 suffices for $\mathbf{a}$42.

The author notes explicitly that for $\mathbf{a}$43 no dichotomy analogous to the rotation case is obtained: the condition $\mathbf{a}$44 is strictly stronger than well-approximability, and whether well-approximability alone forces $\mathbf{a}$45 is left open.

Limitations and open questions

Several qualifications attach to the results above. The spectrum theorem is proved only for alphabets of size at least three; the binary case is posed as an open question, and consequently the density theorem's restriction $\mathbf{a}$46 is provisional. The bounds $\mathbf{a}$47 for Thue–Morse and Rudin–Shapiro are upper bounds, not exact values, and the Rudin–Shapiro proof is only sketched. The conjecture that every non-eventually-periodic overlap-free word has finite $\mathbf{a}$48 remains open, though the paper proves a supporting proposition: for overlap-free $\mathbf{a}$49 with $\mathbf{a}$50, the number of mismatch intervals must satisfy $\mathbf{a}$51, since each maximal run of matches forms a periodic subword of length at most twice its period. Finally, the expected extension of the monomial bracket-word result to all well-approximable $\mathbf{a}$52 is stated as a question rather than a theorem.

Conclusion

The paper establishes that the refined Diophantine exponent attains every value in $\mathbf{a}$53 over ternary alphabets, behaves like the Liouville property in being measure-negligible yet topologically prevalent at each prescribed value, and admits explicit computations across automatic, rotation-coding, and bracket words. Collectively these results delimit the applicability of the Subspace-Theorem transcendence criterion built on $\mathbf{a}$54: the criterion covers a strictly larger class of words than its predecessors, but excludes almost-every word in the Bernoulli sense. The binary spectrum, exact values for automatic words, and the overlap-free and well-approximable-bracket conjectures constitute the natural continuation points identified by the author.