A mathematics preprint dated 20 Aug 2026 reports that the refined Diophantine exponent — a number the paper assigns to an infinite word — has the exact range [1, ∞] over a three-symbol alphabet. An infinite word here is simply an endless sequence of symbols, so the result says there are such sequences with every exponent value at least 1.
The study works with formal infinite words rather than an empirical sample. Its spectrum argument uses a probabilistic method to obtain an infinite binary word with an effective pseudorandomness property.
A complete range, with one typical value
The complete range is an existence statement, not a claim that all values are equally common. Under the uniform Bernoulli product measure on any finite alphabet, the paper states that P(Rdio(a)=1)=1. In plain language, within that specified uniform model, the refined exponent is 1 almost surely.
The almost-sure result is proved with probabilistic arguments based on the Borel–Cantelli lemma.
Density is not the same as probability
That contrast does not eliminate the other values. For alphabets with at least 3 letters, the set of infinite words with any given refined exponent is dense in the Cantor topology. In practical terms, words carrying a selected value can be found arbitrarily close to any target word in the topology used by the study.
The examples pull the exponents apart
The paper also shows a strict separation between Rdio and the related exponent Dio. Over a four-symbol base, for any C>1, there exists an infinite word with Dio(a)<Rdio(a)=C.
For comparison, the Champernowne word has Dio(c)=Rdio(c)=1.
Named sequences give bounds, not final answers
For the Thue–Morse and Rudin–Shapiro sequences, the result is an upper bound rather than an exact value: Rdio is at most 25 in each case. The paper does not determine whether the refined exponent in either sequence actually equals 25.
The Thue–Morse bound is obtained through induction involving base-2 carries to bound mismatches.
Rotations reach the other extreme
Rotation codings produce an opposite extreme. For the Section 7 coding of an irrational rotation using nonconstant piecewise-constant intervals, Rdio(a)=∞.
Continued-fraction arguments give a split for Dio in these rotation codings: it is finite and greater than 1 when the rotation number is badly approximable, and infinite when it is well approximable. In ordinary language, those cases distinguish numbers that resist unusually close rational approximations from numbers that admit them.
Higher-degree cases depend on conditions
For polynomial-coded words of degree at least 2, the theorem concludes that Rdio is finite when the leading coefficient is badly approximable.
Other displayed Diophantine conditions on the coefficients imply Dio(a)=∞. In monomial codings with rational partition boundaries, a stated continued-fraction limsup condition likewise implies Dio(a)=∞.
What the result leaves open
The exact [1, ∞] spectrum has a clear boundary: it is established over a ternary alphabet, while the corresponding binary-alphabet spectrum remains open.
The named-sequence findings remain bounds — 25 or below, not exact evaluations — and the higher-degree polynomial conclusions apply only under their stated approximation assumptions.
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
Original paper · Full text