A Sturmian sequence related to the uniqueness conjecture for Markoff numbers

On a generalization of Christoffel words: epichristoffel words

A characterization of Sturmian sequences by indistinguishable asymptotic pairs

Let $\beta > 1$ be a real number and $x \in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $\beta$-expansion of $x$ ($n \in \mathbb{N}$). It is known that $k_n(x)/n$ converges to $(6\log2\log\beta)/\pi^2$ almost everywhere in the sense of Lebesgue measure. In this paper, we improve this result by proving that the Lebesgue measure of the set of $x \in [0,1)$ for which $k_n(x)/n$ deviates away from $(6\log2\log\beta)/\pi^2$ decays to 0 exponentially as $n$ tends to $\infty$, which generalizes the result of Faivre \cite{lesFai97} from $\beta = 10$ to any $\beta >1$. Moreover, we also discuss which of the $\beta$-expansion and continued fraction expansion yields the better approximations of real numbers.

: One means of bounding the error in digital computation is through the use of interval, or range, arithmetic; instead of computing with approximate real numbers, one calculates with pairs of approximate real numbers -- the first member of a pair being a lower bound for the true result, and the second an upper bound. By this method, one can take into account such varied sources of error as uncertainty in input data, inaccuracies in mathematical formulae, and errors in approximation of real numbers and the operations on them. The theory of interval arithmetic is developed extensively elsewhere. The major obstacle to the use of interval arithmetic is the unavailability of software. INTERVAL is not a standard data type in any production language that we know of; preparation of a package of subprograms to handle interval data is a nontrivial task.

Quantified Constraints Under Perturbation

Best Approximation of Real Numbers by Pisot Numbers

A Metric Result Concerning the Approximation of Real Numbers by Continued Fractions

This chapter offers an overview of words and quadratic numbers, and in particular ordering the conjugates of a Christoffel word. Within this topic the reader learns that the reversal of a Christoffel word is a conjugate, and that the lower and upper Christoffel words of the same slope are the smallest and the largest in their conjugation class. The chapter discusses computation in terms ofMarkoff numbers of the quadratic real number which has a periodic continued fraction with periodic pattern equal to a Christoffel word written on the alphabet 11, 22. It also reviews computation of theMarkoff supremum of a periodic biinfinite sequence, and of theLagrange number of a periodic sequence, both having a periodic pattern as above.

The Markoff equation is the diophantine equation x2 +y2 +z2 = 3xyz. A solution is called a Markoff triple. The main result in this chapter is a bijection between lower Christoffel words and Markoff triples. The bijection uses several ingredients: a special representation of the free monoid into SL2(N), the so-called Fricke relations, which relate the traces of two matrices in SL2, their product and their commutator (an equation reminiscent of the Markoff equation, as noted first by Harvey Cohn). Another lemma describes the socalled Markoff moves: they relate Markoff triples each to another. The chapter ends with a statement of the famous Frobenius conjecture: it asks whether the parametrization of Markoff numbers (that is, components of a Markoff triple), which is surjective by the theorem, is also injective.

We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words by means of this function. Then, we establish a new result on repetitions in Sturmian words and show that it is best possible. Let b ≥ 2 b \ge 2 be an integer. We deduce a lower bound for the irrationality exponent of real numbers whose sequence of b b -ary digits is a Sturmian sequence over { 0 , 1 , … , b − 1 } \{0, 1, \ldots , b-1\} and we prove that this lower bound is best possible. As an application, we derive some information on the b b -ary expansion of log ⁡ ( 1 + 1 a ) \log (1 + \frac {1}{a}) for any integer a ≥ 34 a \ge 34 .

Euclidean strings

CBV functions are computable real functions of bounded variation. In this paper we investigate the basic properties of CBV functions. We are especially interested in the question of whether a real number class is closed under CBV functions. The real number classes considered here include the classes of computable (EC), semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and recursively approximable (RA) real numbers. We show that the classes EC, RA and DBC are closed under CBV functions but SC and WC are not. Furthermore, WC$ is not even closed under computable monotone functions and, finally, the image sets of $\wc$ under computable monotone functions and CBV functions are different.

Gorenstein simplices with a given [formula omitted]-polynomial

Integers without large prime factors

Our goal in this paper is to compute the 2-rank of the class group of the fields K = ℚ ( − 2 , q , d ) \[\mathbb{K}=\mathbb{Q}\left( \sqrt{-2},\sqrt{q},\sqrt{d} \right)\] for any odd positive squarefree integer d and any prime integer q ≡ 5 (mod 8). Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic, of type (2, 2), (2, 4) or (2, 2, 2).