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We carry Sprindžuk’s classification of the complex numbers to the field $\mathbb{Q}_{p}$ of $p$-adic numbers. We establish several estimates for the $p$-adic distance between $p$-adic roots of integer polynomials, which we apply to show that almost all $p$-adic numbers, with respect to the Haar measure, are $p$-adic $\tilde{S}$-numbers of order 1.
We establish several new metrical results on the distribution properties of the sequence ({xn})n≥1, where {·} denotes the fractional part. Many of them are presented in a more general framework, in which the sequence of functions (x ↦ xn)n≥1 is replaced by a sequence (fn)n≥1, under some growth and regularity conditions on the functions fn.
We give transcendence measures for $p$-adic numbers $\unicode[STIX]{x1D709}$, having good rational (respectively, integer) approximations, that force them to be either $p$-adic $S$-numbers or $p$-adic $T$-numbers.
In this paper we prove that the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponent μ in (1/2, 1) is equal to 2(1 − μ) for μ ⩾ $\sqrt2/2$, whereas for μ < $\sqrt2/2$ it is greater than 2(1 − μ) and at most equal to (3 − 2 μ)(1 − μ)/(1 − μ + μ2). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) when μ tends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. Moreover, we prove a lower bound for the packing dimension, which appears to be strictly greater than the Hausdorff dimension for μ ⩾ 0.565. . . .
Let b ⩾ 2 be an integer. Among other results we establish, in a quantitative form, that any sufficiently large integer which is not a multiple of b cannot simultaneously be divisible only by very small primes and have very few nonzero digits in its representation in base b.
Let
$S=\{q_{1},\ldots ,q_{s}\}$
be a finite, non-empty set of distinct prime numbers. For a non-zero integer
$m$
, write
$m=q_{1}^{r_{1}}\cdots q_{s}^{r_{s}}M$
, where
$r_{1},\ldots ,r_{s}$
are non-negative integers and
$M$
is an integer relatively prime to
$q_{1}\cdots q_{s}$
. We define the
$S$
-part
$[m]_{S}$
of
$m$
by
$[m]_{S}:=q_{1}^{r_{1}}\cdots q_{s}^{r_{s}}$
. Let
$(u_{n})_{n\geqslant 0}$
be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every
$\unicode[STIX]{x1D700}>0$
, there exists an integer
$n_{0}$
such that
$[u_{n}]_{S}\leqslant |u_{n}|^{\unicode[STIX]{x1D700}}$
holds for
$n>n_{0}$
. Our proof is ineffective in the sense that it does not give an explicit value for
$n_{0}$
. Under various assumptions on
$(u_{n})_{n\geqslant 0}$
, we also give effective, but weaker, upper bounds for
$[u_{n}]_{S}$
of the form
$|u_{n}|^{1-c}$
, where
$c$
is positive and depends only on
$(u_{n})_{n\geqslant 0}$
and
$S$
.
Let $r\geq 2$ and $s\geq 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the $r$-ary expansion and the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0,1,\ldots ,r-1\}$ and $\{0,1,\ldots ,s-1\}$, and we show that this bound is best possible.
We show how the theory of linear forms in two logarithms allows one to get very good effective irrationality measures for nth roots of rational numbers a/b, when a is very close to b. We give a p-adic analogue of this result under the assumption that a is p-adically very close to b, that is, that a large power of p divides a−b. As an application, we solve completely certain families of Thue–Mahler equations. Our results illustrate, admittedly in a very special situation, the strength of the known estimates for linear forms in logarithms.
Let $\Vert \cdot \Vert$ denote the distance to the nearest integer and, for a prime number $p$, let $|\cdot |_{p}$ denote the $p$-adic absolute value. Over a decade ago, de Mathan and Teulié [Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), 229–245] asked whether $\inf _{q\geqslant 1}$$q\cdot \Vert q{\it\alpha}\Vert \cdot |q|_{p}=0$ holds for every badly approximable real number ${\it\alpha}$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number ${\it\alpha}$ grows too rapidly or too slowly, then their conjecture is true for the pair $({\it\alpha},p)$ with $p$ an arbitrary prime.
Let $b\geq 2$ be an integer and $\hat{v}$ a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers ${\it\xi}$ with the property that, for every sufficiently large integer $N$, there exists an integer $n$ such that $1\leq n\leq N$ and the distance between $b^{n}{\it\xi}$ and its nearest integer is at most equal to $b^{-\hat{v}N}$. We further solve the same question when replacing $b^{n}{\it\xi}$ by $T_{{\it\beta}}^{n}{\it\xi}$, where $T_{{\it\beta}}$ denotes the classical ${\it\beta}$-transformation.
We solve the equation ${x}^{a} + {x}^{b} + 1= {y}^{q} $ in positive integers $x, y, a, b$ and $q$ with $a\gt b$ and $q\geq 2$ coprime to $\phi (x)$. This requires a combination of a variety of techniques from effective Diophantine approximation, including lower bounds for linear forms in complex and $p$-adic logarithms, the hypergeometric method of Thue and Siegel applied $p$-adically, local methods, and the algorithmic resolution of Thue equations.
We prove that if q ≥ 5 is an integer, then every qth power of an integer contains at least 5 nonzero digits in its binary expansion. This is a particular instance of one of a collection of rather more general results, whose proofs follow from a combination of refined lower bounds for linear forms in Archimedean and non-Archimedean logarithms with various local arguments.
In Section 4.2, we have constructed explicitly real numbers which are normal to a given base. In the first section of this chapter, we describe another class of explicit real numbers with the same property. Then, we discuss the existence of explicit examples of absolutely normal numbers.
Definition 5.1. A real number is called absolutely normal if it is normal to every integer base b ≥ 2. A real number is called absolutely non-normal if it is normal to no integer base b ≥ 2.
We briefly and partially mention in Section 5.2 the point of view of complexity and calculability theory. Then, in Section 5.3, we give an explicit example of an absolutely non-normal irrational number. We end this chapter with some words on a method proposed by Bailey and Crandall to investigate the random character of arithmetical constants.
Korobov's and Stoneham's normal numbers
In 1946 Good [324] introduced the so-called ‘normal recurring decimals’. Integers b ≥ 2and k ≥ 1 being given, he constructed rational numbers ξ whose b-ary expansion has period bk and is such that every sequence of k digits from {0, 1,…,b − 1} occurs in the b-ary expansion of ξ with the same frequency b−k. An example with b = 2 and k = 3 is given by the rational 23/255 with purely periodic binary expansion of period 00010111. A similar result was independently proved by de Bruijn [143] also in 1946.
Three of the main questions that motivate the present book are the following:
Is there a transcendental real number α such that ∥αn∥ tends to 0 as n tends to infinity?
Is the sequence of fractional parts {(3/2)n}, n ≥ 1, dense in the unit interval?
What can be said on the digital expansion of an irrational algebraic number?
The latter question amounts to the study of the sequence (ξ10n)n≥1 modulo one, where ξ is an irrational algebraic number. More generally, for given real numbers ξ ≠ 0 and α > 1, we are interested in the distribution of the sequences ({ξαn})n≥1 and (∥ξαn∥)n≥1, where {·} (resp., ∥·∥) denotes the fractional part (resp., the distance to the nearest integer). The situation is very well understood from a metrical point of view. However, for a given pair (ξ, α), our knowledge on ({ξαn})n≥1 is extremely limited, except in very few cases. For instance when ξ = 1 and α is a Pisot number, that is, an algebraic integer (an algebraic integer is an algebraic number whose minimal defining polynomial over ℤ is monic) all of whose Galois conjugates (except itself) are lying in the open unit disc, it is not difficult to show that ∥αn∥ tends to 0 as n tends to infinity.