Soit $f$ une fonction arithmétique. Posons $$m(n;f) \,{\coleq}\, \min_{d|n}\!{\Vert}f(d){\Vert}\qquad(n\,{\geq}\,1),$$ oú ${\Vert} u {\Vert}$ désigne la distance du nombre réel $u$ à l'ensemble des entiers. Le comportement normal de $m$($n$;$f$) est une mesure de la nature probabiliste de l'ensemble des diviseurs d'un entier [Lt ]statistique[Gt ]. Une hypothèse standard d'équirépartition conduit à l'évaluation $$m(n;f) = 1/\tau(n)^{1+o(1)}\quad {\rm pp}$$, oú $\tau$($n$) désigne le nombre total des diviseurs d'un entier naturel $n$, et oú, ici et dans la suite, nous utilisons la mention pp (presque partout) pour indiquer qu'une propriété relative à un entier générique est valable sur une suite de densité naturelle unité.

Let $\Lambda\subseteq \{1,\ldots, N\}$ and let $\{a_{n}\}_{n\in\Lambda}$ be a sequence with $|a_n|\leq 1$ for all $n$. It is easy to see that\[ \left\|\sum_{n\in\Lambda}a_{n}e{(n\theta)}\right\|_{p}\leq \left\|\sum_{n\in\Lambda}e{(n\theta)}\right\|_{p}\] for every even integer $p$. We give an example which shows that this statement can fail rather dramatically when $p$ is not an even integer. This answers in the negative a question known as the Hardy–Littlewood majorant conjecture, thereby ruling out a certain approach to the restriction and Kakeya families of conjectures.

Let $V$ be a complete nonsingular projective surface defined over an algebraic number field $k$, such that the Picard variety of $V$ is trivial and Pic($\bar{V}$) is torsion-free. Since our main interest is in necessary conditions for $V(k)$ not to be empty, we shall further assume that $V(k_v)$ is non-empty for every completion $k_v$ of $k$. We do not assume that $\bar{V}$ is rational, and indeed the case which primarily interests us is when $V$ is a K3 surface. Our objective is to describe effective ways of computing the Brauer–Manin obstructions to the existence of points on $V$ defined over $k$. Write \[ \hbox{Br}^0 (V) = \{\hbox{Ker}(\hbox{Br}(V)\longrightarrow\hbox{Br}(\bar{V}))\}/\{{\rm Im}({\rm Br}(k)\longrightarrow\hbox{Br}(V))\};\] then it is known that $\hbox{Br}^0(V)$ is isomorphic to ${\rm H}^1(k, \hbox{Pic}(\bar{V}))$, though to the best of our knowledge there is in general no known algorithm for computing either $\hbox{Br}^0(V)$ or this isomorphism. It is generally believed that $\hbox{Br}^0(V)$ contains all the information about the Brauer group Br$(V)$ which is useful in this context. Most of this paper is concerned with computing groups isomorphic to $\hbox{Br}^0(V)$, and with describing in terms of these groups the Brauer–Manin obstructions coming from elements of $\hbox{Br}^0(V)$.

In an earlier paper, we solved LeVeque's problem, to establish a central limit theorem for the number of solutions of the diophantine inequality \[ \left\vert x-\frac{p}{q}\right\vert\leq\frac{f(\log q)}{q^2} \] in unknowns $p,q$ with $q\,{>}\,0$, where $f$ is a function satisfying special assumptions and $x$ is chosen randomly in the unit interval. Here, we are interested in the almost sure behavior of the solution set. In particular, we obtain a generalized law of the iterated logarithm and we prove a result that gives strong evidence that the law of the iterated logarithm with the standard norming sequence (suggested by the central limit theorem) holds as well. Both results have to be compared with a theorem of W. M. Schmidt; they imply an inverse to Schmidt's theorem and a strong law of large numbers with an error term that is essentially better than the one provided by Schmidt's result.

A cubic transformation formula for the hypergeometric function $_{2}F_{1}(\frac{1}{3},\frac{2}{3};1;z)$ is proved. As an application of this formula a number of arithmetic convolution sums are evaluated. For example, Melfi's formula, \[ \ukksum{n{-}1}{k{=}1}{k{\equiv}1 \, \mbox{(mod $3$)}}\kern-10pt\sigma(k)\sigma(n-k)=\frac{1}{9}\sigma_3(n),\;\;\;n \equiv 2\, \mbox{(mod $3$)}, \] is proved without the use of modular forms.

The aim of the paper is to provide a criterion for the stability of plane curves using log canonical thresholds. As an application, we compare stable pairs of degree $d$ with stable plane curves with degree $d$.

We give a new proof of Dehn's lemma and the loop theorem. This is a fundamental tool in the topology of 3-manifolds. Dehn's lemma was originally formulated by Dehn, where an incorrect proof was given. A proof was finally given by Papakyriakopolous in his famous 1957 paper where the fundamental idea of towers of coverings was introduced. This was later extended to the loop theorem, and the version used most frequently was given by Stallings.

We have shown that hierarchies for Haken 3-manifolds could be understood by a ‘local version’ of Dehn's lemma and the loop theorem. Developing the idea further enables us here to give a new proof of the classical theorems of Papakyriakopoulos, which do not use towers of coverings. A similar result was obtained by Johannson, with the added assumption that the 3-manifold in question was Haken. Our approach means that no extra hypotheses are necessary. Our method uses the concept of boundary patterns of hierarchies, as developed by Johannson. Marc Lackenby has independently produced a very similar proof in his lecture notes.

Note that the more difficult sphere theorem [4, 10, 25] can then be deduced using Dehn's lemma and the loop theorem, plus the PL theory of minimal surfaces. Other applications, like the important result that a covering of an irreducible 3-manifold is irreducible, then follow also by PL minimal surface theory.

We prove that from a topological point of view most numbers fail to be normal in a spectacular way. For an integer $N \geq 2$ and $x \in [0, 1]$, let $x = \sum^\infty_{n=1} \varepsilon_{N,n}(x)/N^n$, where $\varepsilon_{N, n}(x) \in \{0, 1,\ldots, N - 1\}$ for all $n$, denote the unique non-terminating $N$-adic expansion of $x$. For a positive integer $n$ and a finite string ${\bf i} = i_1\cdots i_k$ with entries $i_j \in \{0, 1,\ldots, N - 1\}$, we write $$ \Pi_N (x,{\bf i};n) = \frac{|\{1 \le i \le n | \varepsilon_{N,i}(x) = i_1, \ldots, \varepsilon_{N,i+k-1} (x) = i_k\}|}{n} $$ for the frequency of the string ${\bf i}$ among the first $n$ digits in the $N$-adic expansion of $x$, and let $\Pi_N^k(x;n) \,{=}\, (\Pi_N(x,{\bf i};n))_{\bf i}$ denote the vector of frequencies $\Pi_N(x,{\bf i};n)$ of all strings ${\bf i} = i_1\cdots i_k$ of length $k$ with entries $i_j \in \{0, 1,\ldots, N-1\}$. We say that a number $x$ is extremely non-normal if each shift invariant probability vector in ${\mathbb R}^{N^k}$ is an accumulation point of the sequence $(\Pi_N^k(x;n))_n$simultaneously for all $k$ and all bases $N$, and we denote the set of extremely non-normal numbers by ${\mathbb E}$, i.e. \begin{eqnarray*} &&{\mathbb E} = \bigcap_N \bigcap_k \big\{x \in [0,1]|\, {\rm each}\, {\bf p} \in \Gamma^k_N\\ &&\quad\hbox{is an accumulation point of the sequence } (\Pi_N (x, {\bf i};n))_n\big\}, \end{eqnarray*} where $\Gamma^k_N$ denotes the simplex of shift invariant probability vectors in ${\mathbb R}^{N^k}$. Our main result says that ${\mathbb E}$ is a residual set, i.e. the complement $[0,1]\backslash {\mathbb E}$ is of the first category. Hence, from a topological point of view, a typical number in [0, 1] is as far away from being normal as possible. This result significantly strengthens results by Maxfield and Schmidt. We also determine the Hausdorff dimension and the packing dimension of ${\mathbb E}$.

Let $E$ be an elliptic curve defined over $\Q$ with complex multiplication by the ring of integers of an imaginary quadratic field $K$. We apply an elliptic unit version of Ichimura–Sumida criterion for the $\Z_p$-extension $F_\infty/F$ associated to $E$ and try to determine the characteristic polynomial of the maximal unramified abelian $p$-extension of $F_\infty$.

Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p\gt 0$. Let $L/K$ be a finite Galois extension with Galois group $G\,{=}\,G_{L/K}$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. By the normal basis theorem, $L$ is always a projective $K[G]$-module. But the ring of integers $\mathcal{O}_L$ is a projective$\mathcal{O}_K[G]$-module if and only $L/K$ is tamely ramified. For wildly ramified extensions, the structure of $\mathcal{O}_L$ as an $\mathcal{O}_K[G]$-module is very complicated and quite far from understood. We propose that the ring of Witt vectors $W\s(\mathcal{O}_L)$ is a more well-behaved object. Indeed, we show that for a large class of extensions $L/K$, the pro-abelian group $H^1(G,W\s(\mathcal{O}_L))$ is zero. We conjecture that this is true in general.

It is known that if every prime power branched cyclic cover of a knot in $S^3$ is a homology sphere, then the knot has vanishing Casson–Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in $S^3$ whose prime power branched cyclic covers are homology spheres. We show that these knots generate an infinite rank subgroup of $\scrf_{(1.0)}/\scrf_{(1.5)}$ for which Casson–Gordon invariants vanish in Cochran–Orr–Teichner's filtration of the classical knot concordance group. As a corollary, it follows that Casson–Gordon invariants are not a complete set of obstructions to a second layer of Whitney disks.

We give a local estimate of the shape of Schottky space at certain boundary points. This is done by studying the geometric limits of sequences of cyclic groups and applying the results to the generators of Schottky groups. We also obtain information on discrete, non-faithful representations close to the cusps.

Given a surface $S$ and an integer $r\,{\ge}\,1$, there is a variety $X_{r-1}$ parametrizing all clusters of $r$ proper and infinitely near points of $S$. We study the geometry of the varieties $X_r$, showing that for every Enriques diagram ${\bf D}$ of $r$ vertices the subset ${\mathit{Cl}}({\bf D})\,{\subset}\,X_{r-1}$ of the clusters with Enriques diagram ${\bf D}$ is locally closed. We study also the relative positions of the subvarieties ${\mathit{Cl}}({\bf D})$, showing that they do not form a stratification and giving criteria for adjacencies between them.

An elliptic space is one whose rational homotopy and rational cohomology are both finite dimensional. David Anick conjectured that any simply connected finite CW-complex $S$ can be realized as the $k$-skeleton of some elliptic complex as long as $k\,{>}\,\dim S$, or, equivalently, that any simply connected finite Postnikov piece $S$ can be realized as the base of a fibration $F\,{\to}\,E\,{\to}\,S$ where $E$ is elliptic and $F$ is $k$-connected, as long as the $k$ is larger than the dimension of any homotopy class of $S$. This conjecture is only known in a few cases, and here we show that in particular if the Postnikov invariants of $S$ are decomposable, then the Anick conjecture holds for $S$. We also relate this conjecture with other finiteness properties of rational spaces.

Let $M$ be a compact, connected surface without boundary and different from $\rp$, and let $B_n(M)$ and $P_n(M)$ denote its braid group and pure braid group on $n$ strings respectively. In this paper, we study the roots of the ‘full twist’ braid in $P_n(M)$ and $B_n(M){\setminus} P_n(M)$. Our main results may be summarised as follows: first, the full twist has no non-trivial root in $P_n(M)$. Further, if $M\ne \St$ and $k\geq 2$, it has a $k\th$ root in $B_n(M){\setminus} P_n(M)$ if and only if $k$ divides either $n$ or $n-1$. This generalises results concerning the sphere of Gillette, Van Buskirk and Murasugi. We also show that the Artin pure braid groups and the pure braid groups of the sphere admit a (non-trivial) splitting as a direct product of which one of the factors is the cyclic group generated by the full twist.

Let $L$ be an unsplittable, prime, oriented, alternating link type in $S^3$. Let $D$ be a reduced alternating diagram representing $L$. We define the Murasugi atoms of $D$ as the oriented link types represented by the prime factors of the Murasugi special components of $D$. We prove (an invariance theorem) that the collection of Murasugi atoms depends only on $L$ and not on $D$. This has the following corollary. Let $L$ be as above and assume that $L$ is achiral. Write its HOMFLY polynomial as $P_{L}(v,z)\,{=}\,\sum_{m}^{M} b_{j}(v) z^j$. Then $b_{M}(v)\,{=}{\pm}\, \beta(v) \beta(v^{-1})$ for some polynomial $\beta(v) \in\mathbb{Z}[v, v^{-1}]$. As a consequence, the leading coefficient of the Conway polynomial of $L$ is a square (up to sign).

The Atiyah–Singer equivariant signature formula implies that the products of isometrically inequivalent classical spherical space forms with the circle are not homeomorphic, and in fact the same conclusion holds if the circle is replaced by a torus of arbitrary dimension. These results are important in the study of group actions on manifolds. Algebraic $K$-theory yields standard classes of counterexamples for topological and smooth analogs of spherical spaceforms. The results of this paper characterize pairs of nonhomeomorphic topological spherical space forms whose products with a given torus of arbitrary dimension are homeomorphic, and the main result is that the known counterexamples are the only ones that exist. In particular, this and basic results in lower algebraic $K$-theory show that if such products are homeomorphic, then the products are already homeomorphic if one uses a 3-dimensional torus. Sharper results are established for important special cases such as fake lens spaces. The methods are basically surgery-theoretic with some input from homotopy theory. One consequence is the existence of new infinite families of manifolds in all dimensions greater than three such that the squares of the manifolds are homeomorphic although the manifolds themselves are not. Analogous results are obtained in the smooth category.

We prove the Martino–Priddy conjecture for an odd prime $p$: the $p$-completions of the classifying spaces of two groups $G$ and $G^\prime$ are homotopy equivalent if and only if there is an isomorphism between their Sylow $p$-subgroups which preserves fusion. A second theorem is a description for odd $p$ of the group of homotopy classes of self homotopy equivalences of the $p$-completion of $BG$, in terms of automorphisms of a Sylow $p$-subgroup of $G$ which preserve fusion in $G$. These are both consequences of a technical algebraic result, which says that for an odd prime $p$ and a finite group $G$, all higher derived functors of the inverse limit vanish for a certain functor $\calz_G$ on the $p$-subgroup orbit category of $G$.

We show that the semi-direct product of a one-ended torsion-free word-hyperbolic group with $\integers$, given through an automorphism $\alpha{:}\, G \,{\to}\, G$, is a hyperbolic group relative to certain canonical subgroups of $G$ on which $\alpha$ acts periodically or with linear growth.

In this paper we consider quaternionic Möbius transformations preserving the unit ball in the quaternions $\bh$. In other words, maps of the form $g(z)=(az+b)(cz+d)^{-1}$ where $a$, $b$, $c$ and $d$ all lie in $\bh$ with the property that $|g(z)|<1$ for all $|z|<1$. We give an explicit expression for the fixed points of $g$ in terms of $a$, $b$, $c$ and $d$ and we use this to classify quaternionic Möbius transformations into six categories determined by their dynamics.