On the complexity of algebraic number I. Expansions in integer bases

Let $b \ge 2$ be an integer. We prove that the $b$-adic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion

Near NP-Completeness for Detecting p-adic Rational Roots in One Variable

We show that deciding whether a sparse univariate polynomial has a p-adic rational root can be done in NP for most inputs. We also prove a polynomial-time upper bound for trinomials with suitably generic p-adic Newton polygon. We thus improve the best previous complexity upper bound of EXPTIME. We also prove an unconditional complexity lower bound of NP-hardness with respect to randomized reductions for general univariate polynomials. The best previous lower bound assumed an unproved hypothesis on the distribution of primes in arithmetic progression. We also discuss how our results complement analogous results over the real numbers.Comment: 8 pages in 2 column format, 1 illustration. Submitted to a conferenc

On the metric theory of numbers in non-Archimedean settings

This thesis is a contribution to some fields of the metrical theory of numbers in non- Archimedean settings. This is a branch of number theory that studies and characterizes sets of numbers, which occur in a locally compact topological field endowed with a non- Archimedean absolute value. This is done from a probabilistic or measure-theoretic point of view. In particular, we develop new formulations of ergodicity and unique ergodicity based on certain subsequences of the natural numbers, called Hartman uniformly distributed sequences. We use subsequence ergodic theory to establish a generalised metrical theory of continued fractions in both the settings of the p-adic numbers and the formal Laurent series over a finite field. We introduce the a-adic van der Corput sequence which significantly generalises the classical van der Corput sequence. We show that it provides a wealth of examples of low-discrepancy sequences which are very useful in the quasi-Monte Carlo method. We use our subsequential characterization of unique ergodicity to solve the generalised version of an open problem asked by O. Strauch on the distribution of the sequence of consecutive van der Corput sequences. In addition to these problems in ergodic methods and number theory, we employ some geometric measure theory to settle the positive characteristic analogue of an open problem asked by R.D. Mauldin on the complexity of the Liouville numbers in the field of formal Laurent series over a finite field by giving a complete characterization of all Hausdorff measures of the set of Liouville numbers