Continued fractions with low complexity: Transcendence measures and quadratic approximation
نویسنده
چکیده
We establish measures of non-quadraticity and transcendence measures for real numbers whose sequence of partial quotients has sublinear block complexity. The main new ingredient is an improvement of Liouville’s inequality giving a lower bound for the distance between two distinct quadratic real numbers. Furthermore, we discuss the gap between Mahler’s exponent w2 and Koksma’s exponent w ∗ 2 .
منابع مشابه
Transcendence measures for continued fractions involving repetitive or symmetric patterns
It was observed long ago (see e.g., [32] or [20], page 62) that Roth’s theorem [28] and its p-adic extension established by Ridout [27] can be used to prove the transcendence of real numbers whose expansion in some integer base contains repetitive patterns. This was properly written only in 1997, by Ferenczi and Mauduit [21], who adopted a point of view from combinatorics on words before applyi...
متن کاملAutomatic Continued Fractions Are Transcendental or Quadratic
We establish new combinatorial transcendence criteria for continued fraction expansions. Let α = [0; a1, a2, . . .] be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients (a`)`≥1 of α is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton. Résumé. Nous établissons de nouveaux critères combinatoires de ...
متن کاملPalindromic continued fractions
An old problem adressed by Khintchin [15] deals with the behaviour of the continued fraction expansion of algebraic real numbers of degree at least three. In particular, it is asked whether such numbers have or not arbitrarily large partial quotients in their continued fraction expansion. Although almost nothing has been proved yet in this direction, some more general speculations are due to La...
متن کاملContinued fractions and transcendental numbers
It is widely believed that the continued fraction expansion of every irrational algebraic number α either is eventually periodic (and we know that this is the case if and only if α is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine in [22] (see also [6,39,41] for surveys including a discussion on this subj...
متن کاملTranscendence with Rosen Continued Fractions
We give the first transcendence results for the Rosen continued fractions. Introduced over half a century ago, these fractions expand real numbers in terms of certain algebraic numbers.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011