Palindromic 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 Lang [16], including the fact that algebraic numbers of degree at least three should behave like most of the numbers with respect to the Gauss–Khintchin–Kuzmin–Lévy laws. A preliminary step consists in providing explicit examples of transcendental continued fractions. The first result of this type is due to Liouville [17], who constructed real numbers whose sequence of partial quotients grows very fast, too fast for being algebraic. Subsequently, various authors used deeper transcendence criteria from Diophantine approximation to construct other classes of transcendental continued fractions. Of particular interest is the work of Maillet [18] (see also Section 34 of Perron [19]), who was the first to give examples of transcendental continued fractions with bounded partial quotients. Further examples were provided by A. Baker [8, 9], Davison [11], Queffélec [20], Allouche et al. [7], Adamczewski and Bugeaud [1, 5], and Adamczewski et al. [6], among others. A common feature of all these results is that they apply to real numbers whose continued fraction expansion is ‘quasi-periodic’ in the sense that it contains arbitrarily long blocks of partial quotients which occur precociously at least twice. Continued fractions beginning with arbitrarily large palindromes appear in several recent papers [21, 22, 10, 13, 2]. Motivated by this and the general problematic mentioned above, we ask whether precocious occurrences of some symmetric patterns in the continued fraction expansion of an irrational real number do imply that the latter is either quadratic, or transcendental. We obtain three new transendence criteria that apply to a broad class of continued fraction expansions, including expansions with unbounded partial quotients. These results provide the exact counterpart of [1] (see also Theorem 3.1 from [6]), with periodic patterns being replaced by symmetric ones. Like in [1], their proofs heavily depend on the Schmidt Subspace Theorem [24]. As already mentioned, there is a long tradition in using an excess of periodicity to prove the transcendence of some continued fractions. This is indeed very natural: if the continued fraction expansion of the real number ξ begins with, say, the periodic pattern ABBB (here, A, B denote two finite blocks of partial quotients), then ξ is ‘very close’ to the quadratic irrational real number having the eventually periodic