Transcendence and measures via the refined Diophantine exponent

Technical Summary: Transcendence and Measures via the Refined Diophantine Exponent

Problem Statement
The paper addresses the Diophantine properties of real numbers $\xi = \sum_{i \ge 0} a_i \beta^{-i}$, where the coefficients $a_i$ form an infinite word over a finite set of integers (or $S$-integers) and $\beta$ is an algebraic number with $|\beta| > 1$. The central problem is determining the transcendence of $\xi$ and establishing quantitative transcendence measures (bounds on how well $\xi$ can be approximated by algebraic numbers).

Classical approaches, notably those by Adamczewski and Bugeaud, utilize the Diophantine exponent ($Dio(a)$) to measure the periodicity of the coefficient word $a$. If $Dio(a)$ is sufficiently large relative to the height of $\beta$, $\xi$ is either algebraic (of a restricted form) or transcendental. However, the classical exponent requires exact repetition of subwords. This rigidity limits its applicability to words where repetition is "obscured" by noise or where the repetition is only approximate (e.g., in certain dynamical systems or words with specific combinatorial structures like lacunary sequences). While previous works (Corvaja-Zannier, Bell-Diller-Jonsson, Kebis-Luca-Ouaknine-Scoones-Worrell) have addressed specific cases using higher-dimensional Subspace Theorem applications, a unified framework capable of handling "noisy" repetition while preserving the strength of the classical dichotomy was lacking.

Methodology
The author introduces a new combinatorial invariant, the Refined Diophantine Exponent ($Rdio(a)$), which generalizes the classical $Dio(a)$.

1. Definition of $(\epsilon, \delta)$-closedness: Instead of requiring two subwords to be identical, the new definition allows them to differ on a set of indices that can be covered by $\delta$ intervals with total length at most $\epsilon$ times the word length. This models "noise" or mismatches.
2. Refined Exponent: $Rdio(a)$ is the supremum of $\rho$ such that the word $a$ contains subwords satisfying this approximate repetition condition with a growth rate $t_n \ge \rho s_n$.
3. Subspace Theorem Application:
   • Qualitative (Theorem A): The proof adapts the $p$-adic Subspace Theorem. The "mismatches" (noise) are treated as additional variables in the linear forms. By constructing a system of linear forms involving $\xi$, its rational approximations, and the mismatch terms, the author shows that if $\xi$ is algebraic, the coefficients of the mismatch terms must vanish, forcing $\xi$ into a restricted algebraic form.
   • Quantitative (Theorem B): To derive transcendence measures, the author employs Evertse's Quantitative Subspace Theorem. This requires controlling the growth of the sequences $(r_n, s_n, t_n)$ and the height of the approximations. The proof involves constructing dense rational approximations and bounding the number of solutions to the resulting inequalities, leading to explicit lower bounds for $|\xi - \alpha|_v$.
4. Mahler's Classification: The results are interpreted within Mahler's classification of transcendental numbers ($A, S, T, U$-numbers), distinguishing between numbers with finite or infinite approximation exponents.

Key Contributions and Results

• Theorem A (Transcendence Dichotomy): If $Rdio(a) > \frac{\log H(\beta)}{\log |\beta|_v}$, then $\xi$ is either in the number field $K$ (and of a very restricted form) or transcendental. This unifies and extends previous results by Adamczewski-Bugeaud, Corvaja-Zannier, and others, showing that "noise" in the repetition does not prevent transcendence detection.
• Theorem B (Transcendence Measure): Under stronger asymptotic conditions on the sequences (Condition $(**)_\rho$), the paper establishes a quantitative lower bound for $|\xi - \alpha|_v$. Specifically, for algebraic $\alpha \notin K$ of degree $d$ and height $H(\alpha)$, $|\xi - \alpha|_v \ge H(\alpha)^{-(2d)^c (\log 4d)(\log \log 4d)}$. This implies $\xi$ is either a $U_d$-number (for small $d$) or an $S$- or $T$-number.
• Theorem C (Lacunary Numbers): For lacunary series $\xi = \sum a_i \beta^{-u_i}$, the paper provides a dichotomy based on the growth of the gaps $u_{i+1}/u_i$. If the gaps grow super-exponentially, $\xi$ is a $U_d$-number; otherwise, it is an $S$- or $T$-number. This improves upon previous bounds by removing the requirement that coefficients be $S$-integers.
• Theorem D (Sturmian and $k$-bonacci Words): For $\beta$ being a Pisot number of degree at most 2, the paper proves that numbers generated by Sturmian words (with bounded partial quotients) and $k$-bonacci words are $S$- or $T$-numbers. This is achieved by combining the refined exponent results with dense rational approximations constructed via the properties of these words.

Significance and Claims
The paper claims to provide a unified framework that recovers and extends existing transcendence results for various classes of infinite words (lacunary, stuttering, echoing, and those from algebraic dynamics).

• Robustness: The primary significance is the ability to handle "noisy" repetition. The refined exponent $Rdio(a)$ detects periodicity even when subwords are not identical, a scenario where the classical $Dio(a)$ might fail (e.g., $Dio(a)=1$ while $Rdio(a)=\infty$).
• Quantitative Refinement: The work moves beyond qualitative transcendence to provide explicit transcendence measures, classifying the resulting numbers within Mahler's hierarchy.
• Connection to Dynamics: The methodology is motivated by and connects to recent breakthroughs in algebraic dynamics (Bell, Diller, Jonsson), specifically regarding the transcendence of dynamical degrees.
• Limitations: The author modestly notes that while the method yields weaker results than the classical Padé approximant approach (which requires functional equations), it applies to uncountable families of functions and is robust under perturbations. Furthermore, while the method establishes that numbers are not $U_1$ or $U_2$ in certain cases, determining whether they are $U_d$ for higher $d$ remains difficult when the classical Diophantine exponent is finite.

In summary, the paper advances the field by generalizing the combinatorial criteria for transcendence to include approximate repetitions, thereby broadening the scope of numbers for which transcendence and transcendence measures can be effectively established.