By Yann Bugeaud

This e-book offers state of the art examine at the distribution modulo one in all sequences of quintessential powers of actual numbers and similar themes. many of the effects have by no means prior to seemed in a single booklet and lots of of them have been proved merely over the last decade. subject matters coated comprise the distribution modulo one of many quintessential powers of 3/2 and the frequency of incidence of every digit within the decimal enlargement of the sq. root of 2. the writer takes some extent of view from combinatorics on phrases and introduces quite a few options, together with specific structures of standard numbers, Schmidt's video games, Riesz product measures and transcendence effects. With various routines, the booklet is perfect for graduate classes on Diophantine approximation or as an advent to distribution modulo one for non-experts. experts will take pleasure in the inclusion of over 50 open difficulties and the wealthy and finished bibliography of over seven hundred references.

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**Sample text**

Hence, β −1 , the complex conjugate of β, is also a root of the minimal deﬁning polynomial Q(X) of α over Z. This shows that Q(X) = X d Q(1/X), where d is the degree of Q(X), and proves the claim. Pisot [564] used the terminology ‘nombre ρ’ (literally, ρ-number) to denote algebraic integers whose Galois conjugates are all in the open unit disc. Vijayaraghavan [708] denoted by S the set of these numbers and proved that S contains algebraic integers of every degree, a result which was independently established in [686].

Let qd X d + · · · + q1 X + q0 be the minimal deﬁning polynomial of α over the integers. Let ε be a positive real number with 0 < ε < 1/(|q0 | + · · · + |qd |). Assume that the sequence ({ξαn })n≥1 has only ﬁnitely many limit points, denoted by ζ1 , . . , ζr . 18, there exists a positive integer q such that ||qζi || < ε/2 for i = 1, . . , r. Consequently, there exists n0 such that, for every n ≥ n0 , we have ||qξαn || ≤ ε. Let n ≥ n0 be an integer. Write qξαn = an + ε n , where an is a rational integer and |ε n | ≤ ε.

Bt be positive integers. If there exist two sequences (mk )k≥1 and (hk )k≥1 of positive integers with (hk )k≥1 being unbounded and, for k ≥ 1, amk +j+nt = bj , for j = 1, . . , t and n = 0, . . , hk − 1, then inf q · qξ · |q|p = 0 q≥1 for every prime number p. 34) holds for every prime p and every quadratic number ξ, a result already proved in [483]. Einsiedler and Kleinbock [277] established that the set of possible exceptions to the de Mathan–Teuli´e conjecture is, from the metric point of view, very small.