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Understanding Regular Episturmian Words and Diophantine Exponents


Core Concepts
The author explores the complexity of regular episturmian words and their Diophantine exponents, providing insights into their unique properties and relationships.
Abstract
Regular episturmian words are analyzed for their initial nonrepetitive complexity and Diophantine exponents. The study reveals novel results on irrationality exponents, transcendental numbers, and Liouville numbers. The concept of Ostrowski numeration systems is introduced to represent these words efficiently. The intercept of episturmian words is defined based on sequences of finite words, leading to a deeper understanding of their structure. Theorems are presented to characterize the properties of regular episturmian words and their numerical representations. Key results include the relationship between partial quotients, greedy expansions, and Ostrowski expansions in the context of regular directive words. Theorems demonstrate how these concepts simplify the analysis of complex word structures. Overall, the study sheds light on the intricate nature of regular episturmian words and provides a framework for efficient representation and analysis using generalized standard words.
Stats
We present a method to evaluate the initial nonrepetitive complexity of regular episturmian words extending the work of Wojcik on Sturmian words. We prove that the Diophantine exponent of a regular episturmian word is finite if its directive word has bounded partial quotients. The sequence x1x2 · · · equals the periodic sequence with period 012 · · · (d − 1) for some d.
Quotes

Deeper Inquiries

How do regular episturmian words compare to other types of Sturmian or Fibonacci-like sequences

Regular episturmian words, which are a subclass of episturmian words with directive words of a specific form, can be compared to other types of Sturmian or Fibonacci-like sequences in terms of their properties and characteristics. Sturmian Words: Regular episturmian words share similarities with Sturmian words as they both have restricted forms that exhibit unique combinatorial properties. However, regular episturmian words have directive words with a periodic structure (such as the Fibonacci word) beyond binary alphabets. Fibonacci-Like Sequences: While regular episturmian words include d-bonacci sequences like the Fibonacci and Tribonacci sequences, they extend beyond these well-known examples by allowing for more general periodic structures in their directive words. In essence, regular episturmian words bridge the gap between traditional Sturmian and Fibonacci-like sequences by incorporating periodicity into their construction while maintaining the essential properties that define these types of word sequences.

What implications do these findings have for number theory beyond just word combinatorics

The findings related to regular episturmian words have significant implications for number theory beyond just word combinatorics: Diophantine Exponents: By characterizing the initial nonrepetitive complexity and Diophantine exponents of regular episturnmiamnwords, novel results on irrationality exponents are obtained. This provides lower bounds for irrationality exponents of real numbers whose fractional parts match regular episturnmiamnwords. Transcendental Numbers: The identification of a new uncountable class of transcendental numbers whose irrationality exponents are strictly greater than 2 expands our understanding of transcendental numbers and their properties. These implications deepen our knowledge not only in combinatorics but also in number theory by establishing connections between word structures and mathematical constants.

How can the concept of Ostrowski numeration systems be applied in other mathematical contexts

The concept of Ostrowski numeration systems introduced in the context above can be applied in various mathematical contexts outside word combinatorics: Number Theory: Ostrowski numeration systems provide an alternative way to represent integers using different bases derived from certain integer alphabets. These representations can offer insights into number theoretic properties such as factorization patterns or divisibility rules based on the chosen base system. Algorithms: In algorithm design, utilizing Ostrowski numeration systems can lead to efficient algorithms for arithmetic operations or encoding schemes where unconventional numeral representations may offer advantages over traditional positional notation systems. Cryptography: The unique properties of Ostrowski numeration systems could potentially be leveraged in cryptographic protocols or encryption techniques where non-standard numeral representations add an extra layer of security through obfuscation methods based on these alternative bases. By exploring applications beyond word combinatorics, Ostrowski numeration systems open up avenues for innovative approaches across various mathematical domains.
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