Let α be a real number and k a positive integer. We shall be interested in integer values of n for which ║αnk║ is small. For the case k = 1 we have Dirichlet's Theorem. For any N ≥ 1 there exists n ≤ N with

On Waring's problem for cubes, it is conjectured that every sufficiently large natural number can be represented as a sum of four cubes of natural numbers. Denoting by E(N) the number of the natural numbers up to N that cannot be written as a sum of four cubes, we may express the conjecture as E(N)≪1.

The following theorem has been proved by Bambah, Rogers and Zassenhaus [1],

Theorem A. Let K be a closed convex domain with a centre. Let

be points such that:

(i) the polygon A0 A1 … An is a Jordan polygon bounding a closed domain π of area a(π);

(ii) for each r, 0 ≤ r ≤ n, there is a point common to K + An-1 and K + Ar;

(iii) the points An+1, …. An+m are in the interior of π;

(iv) for each point X of π, there exists an Ar, 1 ≤ r ≤ n + m, such that X ∈ K + Ar and the line segment XAr is in π. Then

where t(K) is the area of the largest triangle contained in K.

Let G be a finite connected graph with no loops or multiple edges. The point-connectivity K(G) of G is the minimum number of points whose removal results in a disconnected or trivial graph. Similarly, the line-connectivity λ(G) of G is the minimum number of lines whose removal results in a disconnected or trivial graph. For the complete graph Kp we have

We attempt to develop a new chapter of the theory of uniform distribution; we call it strong uniformity. Strong uniformity in a nutshell means that we combine Lebesgue measure with the classical theory of uniform distribution, basically founded by Weyl in his famous paper from 1916 [Über die Gleichverteilung von Zahlen mod Eins, Math. Ann.77 (1916), 313–352], which is built around nice test sets, such as axis-parallel rectangles and boxes. We prove the continuous version of the well-known Khinchin’s conjecture [Eins Satz über Kettenbrüche mit arithmetischen Adwendungen, Math. Z.18 (1923), 289–306] in every dimension $d\geqslant 2$ (the discrete version turned out to be false—it was disproved by Marstrand [On Khinchin’s conjecture about strong uniform distribution, Proc. Lond. Math. Soc. (3) 21 (1970), 540–556]). We consider an arbitrarily complicated but fixed measurable test set $S$ in the $d$-dimensional unit cube, and study the uniformity of a typical member of some natural families of curves, such as all torus lines or billiard paths starting from the origin, with respect to $S$. In the two-dimensional case we have the very surprising superuniformity of the typical torus lines and billiard paths. In dimensions ${\geqslant}3$ we still have strong uniformity, but not superuniformity. However, in dimension three we have the even more striking super-duper uniformity for two-dimensional rays (replacing the torus lines). Finally, we indicate how to exhibit superuniform motions on every “reasonable” plane region (e.g., the circular disk) and on every “reasonable” closed surface (sphere, torus and so on).

Let $n\geqslant C$ for a large universal constant $C>0$ and let $B$ be a convex body in $\mathbb{R}^{n}$ such that for any $(x_{1},x_{2},\ldots ,x_{n})\in B$, any choice of signs $\unicode[STIX]{x1D700}_{1},\unicode[STIX]{x1D700}_{2},\ldots ,\unicode[STIX]{x1D700}_{n}\in \{-1,1\}$ and for any permutation $\unicode[STIX]{x1D70E}$ on $n$ elements, we have $(\unicode[STIX]{x1D700}_{1}x_{\unicode[STIX]{x1D70E}(1)},\unicode[STIX]{x1D700}_{2}x_{\unicode[STIX]{x1D70E}(2)},\ldots ,\unicode[STIX]{x1D700}_{n}x_{\unicode[STIX]{x1D70E}(n)})\in B$. We show that if $B$ is not a cube, then $B$ can be illuminated by strictly less than $2^{n}$ sources of light. This confirms the Hadwiger–Gohberg–Markus illumination conjecture for unit balls of $1$-symmetric norms in $\mathbb{R}^{n}$ for all sufficiently large $n$.

We prove an asymptotic formula for squarefree numbers in arithmetic progressions, improving previous results by Prachar and Hooley. As a consequence we improve a lower bound of Heath-Brown for the least squarefree number in an arithmetic progression.

For a hyperbolic surface, embedded eigenvalues of the Laplace operator are unstable and tend to dissolve into scattering poles i.e. become resonances. A sufficient dissolving condition was identified by Phillips–Sarnak and is elegantly expressed in Fermi’s golden rule. We prove formulas for higher approximations and obtain necessary and sufficient conditions for dissolving a cusp form with eigenfunction $u_j$ into a resonance. In the framework of perturbations in character varieties, we relate the result to the special values of the $L$-series $L(u_j\otimes F^n, s)$. This is the Rankin–Selberg convolution of $u_j$ with $F(z)^n$, where $F(z)$is the antiderivative of a weight two cusp form. In an example we show that the above-mentioned conditions force the embedded eigenvalue to become a resonance in a punctured neighborhood of the deformation space.

We construct a polyhedron with ten vertices of genus three which has three axes of symmetry. It is as symmetric as possible. Ten is the minimal number of vertices which a polyhedron of genus three can have. A modification of our polyhedron yields a symmetric polyhedral realization of Dyck's regular map.

This paper deals with diophantine equations and inequalities in three variables x, y, z. Write

Given an algebraic number field K of degree n and an element ξ of K, denote the conjugates of ξ by ξ(1) = ξ, ξ(2), …, ξ(n), and its norm ξ(1) … ξ(n) by N(ξ). Recent results [3] in diophantine approximations enable us to prove the following theorems.

In a typical counter-example construction in geometric measure theory, starting from some initial set one obtains by successive reductions a decreasing sequence of sets Fn, whose intersection has some required property; it is desired that ∩ Fn shall have large Hausdorf F dimension. It has long been known that this can often be accomplished by making each Fn+1 sufficiently “dense” in Fn. Our first theorem expresses this intuitive idea in a precise form that we believe to be both new and potentially useful, if only for simplifying the exposition in such cases. Our second theorem uses just such a construction to solve the problem that originally stimulated this work: can a Borel set in ℝk have Hausdorff dimension k and yet for continuum-many directions in every angle have at most one point on each line in that direction? The set of such directions must have measure zero, since in fact in almost all directions there are lines that meet the Borel set (of dimension k) in a set of dimension 1: this can easily be deduced from Theorem 6.6 of Mattila [5], which generalized Marstrand's result [4] for the case k = 2.

§1. Introduction. We consider the spectral function ρα(λ) associated with the Sturm–Liouville equation

with the boundary condition

Let N(ρ; ω) be the number of points of a d-dimensional lattice Γ. where d≥2, inside a ball of radius ρ centred at the point ω. Denote by (ρ) the number N(ρ; ω) averaged over ω in the elementary cell Ω of the lattice Γ. The main result is the following lower bound for for dimensions d ≅ l(mod 4):

The statement from the title is proved.

Let Ω be a convex planar domain, with no curvature or regularity assumption on the boundary. Let Nθ(R) = card{RΩθ∩ℤ2}, where Ωθ denotes the rotation of Ω by θ. It is proved that, up to a small logarithmic transgression, Nθ(R) = |Ω|R2 + O(R2/3), for almost every rotation. A refined result based on the fractal structure of the image of the boundary of Ω under the Gauss map is also obtained.

A general convex body in Euclidean space can be approximated by smooth convex bodies, and many results, arising first in the differential geometry of smooth convex bodies, have been extended to yield corresponding results for general convex bodies. Although convex bodies can be approximated by convex polyhedra, very little of the rich theory of convex polyhedra has been extended to general convex bodies. In this paper, we extend the concept of the one-skeleton of a convex polytope to yield the concept of the one-skeleton of a general convex body. We investigate the connectivity properties of this one-skeleton, and we extend a result of Balinski [1], on paths in the one-skeleton of a convex polytope to the class of convex bodies.

In a recent paper of Ellenberg, Oberlin, and Tao [The Kakeya set and maximal conjectures for algebraic varieties over finite fields. Mathematika 56 (2010), 1–25], the authors asked whether there are Besicovitch phenomena in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{n} $. In this paper, we answer their question in the affirmative by explicitly constructing a Kakeya set of measure zero in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{n} $. Furthermore, we prove that any Kakeya set in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{2} $ or ${ \mathbb{Z} }_{p}^{2} $ is of Minkowski dimension 2.

It is well known that a tournament (complete oriented graph) on n vertices has at most directed triangles, and that the constant is best possible. Motivated by some geometric considerations, our aim in this paper is to consider some “higher order” versions of this statement. For example, if we give each 3-set from an n-set a cyclic ordering, then what is the greatest number of “directed 4-sets” we can have? We give an asymptotically best possible answer to this question, and give bounds in the general case when we orient each d-set from an n-set.

Given $n\in \mathbb{N}$ and $\unicode[STIX]{x1D70F}>1/n$, let ${\mathcal{S}}_{n}(\unicode[STIX]{x1D70F})$ denote the classical set of $\unicode[STIX]{x1D70F}$-approximable points in $\mathbb{R}^{n}$, which consists of $\mathbf{x}\in \mathbb{R}^{n}$ that lie within distance $q^{-\unicode[STIX]{x1D70F}-1}$ from the lattice $(1/q)\mathbb{Z}^{n}$ for infinitely many $q\in \mathbb{N}$. In pioneering work, Kleinbock and Margulis showed that for any non-degenerate submanifold ${\mathcal{M}}$ of $\mathbb{R}^{n}$ and any $\unicode[STIX]{x1D70F}>1/n$ almost all points on ${\mathcal{M}}$ are not $\unicode[STIX]{x1D70F}$-approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the Hausdorff measure and dimension of the associated null set ${\mathcal{M}}\cap {\mathcal{S}}_{n}(\unicode[STIX]{x1D70F})$. In this paper we suggest a new approach based on the Mass Transference Principle of Beresnevich and Velani [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], which enables us to find a sharp lower bound for $\dim {\mathcal{M}}\cap {\mathcal{S}}_{n}(\unicode[STIX]{x1D70F})$ for any $C^{2}$ submanifold ${\mathcal{M}}$ of $\mathbb{R}^{n}$ and any $\unicode[STIX]{x1D70F}$ satisfying $1/n\leqslant \unicode[STIX]{x1D70F}<1/m$. Here $m$ is the codimension of ${\mathcal{M}}$. We also show that the condition on $\unicode[STIX]{x1D70F}$ is best possible and extend the result to general approximating functions.

We obtain explicit bounds on the moments of character sums, refining estimates of Montgomery and Vaughan. As an application we obtain results on the distribution of the maximal magnitude of character sums normalized by the square root of the modulus, finding almost double exponential decay in the tail of this distribution.