A mathematical preprint reports that the refined Diophantine exponent, a measure built around repetition in an infinite word, can take every value from 1 through infinity when the word uses a three-symbol alphabet. That gives the study a full ternary spectrum: no level in that range is missing. The paper studies infinite words over finite alphabets and defines the refined exponent as the supremum of the real numbers for which a word satisfies a repetition condition that allows mismatches.
A scale for repetition
Put plainly, the exponent asks how high a repetition threshold an infinite word can meet while still passing the paper's mismatch-tolerant test. The word "supremum" means the paper is identifying the upper edge of the values that satisfy that condition, rather than simply counting exact repeated blocks. This gives the results a single scale for comparing otherwise different infinite-word constructions.
One typical value, many possible levels
That complete range sits alongside a sharply different probability result. Under uniform Bernoulli measure, the refined exponent equals 1 almost surely for every finite alphabet with at least two symbols. For alphabets with at least three symbols, every level from 1 through infinity is dense in the Cantor topology. In other words, the model's probability-one result singles out 1, while the topological result says that all allowed levels remain densely present.
The construction behind the spectrum uses the probabilistic method and uniform Bernoulli measure to establish the existence of a suitably pseudorandom binary word. The proof of the almost-sure statement invokes the Borel-Cantelli lemma. In this way, the paper joins an existence argument for the full range to a measure-theoretic result about the value seen almost surely.
Examples do not all agree
The paper then turns to explicit sequences. The Champernowne word has both its refined and ordinary Diophantine exponents equal to 1. For the Thue-Morse and Rudin-Shapiro sequences, the refined exponent is shown to be no greater than 25. These last two findings are upper bounds, not exact values, so they set a ceiling without saying where the sequences fall below it.
Rotation-coded words show a different split. For the specified nonconstant coding of an irrational rotation by intervals, the refined exponent is infinite. The ordinary exponent is finite and greater than 1 for badly approximable rotation numbers, but infinite for well approximable rotation numbers. The paper therefore reports different behavior for the refined and ordinary exponents within this same type of construction.
Conditional patterns in bracket words
Further conditional results come from polynomial bracket words. In the higher-degree construction, a badly approximable leading coefficient implies a finite refined exponent. Under the stated coefficient approximation condition, the ordinary exponent is infinite. For the monomial bracket-word construction, the stated unbounded continued-fraction denominator condition likewise implies an infinite ordinary exponent. Each of these conclusions is tied to the approximation assumption named for its construction.
Publication status
The document is an arXiv preprint, version 2, dated 5 October 2026. The project reports funding from the European Union's MSCA-Horizon Europe programme under grant agreement No. 101126554.
Paper data and sources
Original title: Spectrum of the refined Diophantine exponent
Authors: Quang-Khai Nguyen
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
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