Number Theory and Formal Languages

Shallit, Jeffrey

doi:10.1007/978-1-4612-1544-8_22

Cited by 9 publications

(5 citation statements)

References 57 publications

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“…Our main result gives a positive answer to a conjecture of Shallit [32]. Actually, we will prove a quantitative version of Theorem 2.1.…”

Section: Resultsmentioning

confidence: 69%

“…This uses some of the ideas in [3,6] together with classical techniques from Diophantine approximation and a careful combinatorial study of automatic sequences. In particular, we will prove that no Liouville number can be generated by a finite automaton; a result conjectured by Shallit [32]. Actually our approach is much more precise and it will provide an explicit general upper bound for the irrationality measure of any automatic real number.…”

Section: Introductionmentioning

confidence: 87%

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Diophantine properties of real numbers generated by finite automata

Adamczewski¹,

Cassaigne²

2006

Compositio Math.

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“…Our main result gives a positive answer to a conjecture of Shallit [32]. Actually, we will prove a quantitative version of Theorem 2.1.…”

Section: Resultsmentioning

confidence: 69%

Section: Introductionmentioning

confidence: 87%

Diophantine properties of real numbers generated by finite automata

Adamczewski¹,

Cassaigne²

2006

Compositio Math.

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“…Roth's theorem built on work of Liouville [Lio44], who showed that if ξ is an algebraic number of degree d over Q, then µ(ξ) ≤ d. Using this fact, Liouville produced the first examples of transcendental numbers by constructing real numbers with infinite irrationality exponent; numbers with infinite irrationality exponent are now called Liouville numbers in his honour. Towards classifying irrationality exponents of automatic numbers as well as settling a conjecture of Shallit [Sha99], Adamczewski and Cassaigne [AC06] proved that a Liouville number cannot be generated by a finite automaton.…”

mentioning

confidence: 99%

Diophantine approximation of Mahler numbers

Bell

Bugeaud

Coons

2015

Proceedings of the London Mathematical Society

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Abstract. Suppose that F (x) ∈ Z [[x]] is a Mahler function and that 1/b is in the radius of convergence of F (x) for an integer b ≥ 2. In this paper, we consider the approximation of F (1/b) by algebraic numbers. In particular, we prove that F (1/b) cannot be a Liouville number. If, in addition, F (x) is regular, we show that F (1/b) is either rational or transcendental, and in the latter case that F (1/b) is an S-number or a T -number in Mahler's classification of real numbers.

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“…Finally we propose the following problem related to the discussion after Theorem 7.5. This is an analogue of a conjecture of Shallit [34] that was settled in [5]. Problem 8.9.…”

Section: Diophantine Exponents Of D-bonacci Wordsmentioning

confidence: 87%