Using the polynomial method of Dvir [On the size of Kakeya sets in finite fields. Preprint], we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to algebraic varieties W over finite fields F. For instance, given an (n−1)-dimensional projective variety W⊂¶n(F), we establish the Kakeya maximal estimate for all functions f:Fn→R and d≥1, where for each w∈W, the supremum is over all irreducible algebraic curves in Fn of degree at most d that pass through w but do not lie in W, and with Cn,W,d depending only on n,d and the degree of W; the special case when W is the hyperplane at infinity in particular establishes the Kakeya maximal function conjecture in finite fields, which in turn strengthens the results of Dvir.

Let A be an n×m matrix with real entries. Consider the set BadA of x∈[0,1)n for which there exists a constant c(x)>0 such that for any q∈ℤm the distance between x and the point {Aq} is at least c(x)|q|−m/n. It is shown that the intersection of BadA with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.

This paper extends Hlawka’s theorem (from the point of view of Siegel and Weil) on SL(n,ℝ)/SL(n,ℤ) to Sp(n,ℝ)/Sp(n,ℤ). Namely, if Vn=vol(Sp(n,ℝ)/Sp(n,ℤ), where the measure is the Sp(n,ℝ)-invariant measure on Sp(n,ℝ)/Sp(n,ℤ), then Vn can be expressed in terms of the Riemann zeta function by As a consequence, let D be a domain of a sufficiently regular set in ℝ2n. Then:

1. (i) if vol(D)>Vn, then some lattice in ℝ2n contains a non-zero point of D;

2. (ii) if vol(D)<Vn, then some lattice in ℝ2n contains only the zero point of D;

3. (iii) if D is star-shaped about the origin and vol(D)<ζ(2n)Vn, then some lattice in ℝ2n contains only the zero point of D.

It is shown here that given a discrete (and infinite) set of points in the plane, it is possible to arrange them on a polygonal path so that every angle on the polygonal path is at least 9∘. This has been known to hold for finite sets (with 20∘). The main result holds for discrete sets in higher dimensions as well, with a smaller bound on the angle.

It is shown that a separable Hilbert space can be covered by non-overlapping closed convex sets Ci with outer radii uniformly bounded from above and inner radii uniformly bounded from below. This answers a question originating from the work of Klee.

We obtain the formula for the twisted harmonic second moment of the L-functions associated with primitive Hecke eigenforms of weight 2. A consequence of our mean-value theorem is reminiscent of recent results of Conrey and Young on the reciprocity formula for the twisted second moment of Dirichlet L-functions.

We prove two results about the asymptotic formula for The first result is for x7/12+ε≤h≤x and h/(log x)B≤Q≤h, where ε,B>0 are arbitrary constants. For the second result we assume that the Generalized Riemann Hypothesis holds and we obtain a stronger error term and a better uniformity on h.

Given some unit vectors a1,…,am∈ℝn that span all ℝn and some positive numbers θ1,…,θm, we consider for every p≥1 the convex body We will give some upper bounds for the volume of Kp and some lower bounds for the volume of its polar, depending on some parameters, which improve the ones obtained using the Brascamp–Lieb inequality. We will also see how the best choice of these parameters is related to the transformation which takes Kp to a special position which, for instance, when p=∞, is John’s position.

This article considers a family of functionals J to be maximized over the planar convex sets K for which the perimeter and Steiner point have been fixed. Assuming that J is the integral of a positive quadratic expression in the support function h and its derivative, the maximizer is always either a triangle or a line segment (which can be considered as a collapsed triangle). Among the concrete consequences of the main theorem is the fact that, given any convex body K1 of finite perimeter, the set in this class that is farthest away in the sense of the L2 distance is always a line segment. The same property is proved for the Hausdorff distance.

We find the almost-sure Hausdorff and box-counting dimensions of random subsets of self-affine fractals obtained by selecting subsets at each stage of the hierarchical construction in a statistically self-similar manner.

The Hausdorff dimension and measure of the set of simultaneously ψ-approximable points lying on integer polynomial curves is obtained for sufficiently small error functions.

The covariogram of a compact set A⊂ℝn is the function that to each x∈ℝn associates the volume of A∩(A+x). Recently it has been proved that the covariogram determines any planar convex body, in the class of all convex bodies. We extend the class of sets in which a planar convex body is determined by its covariogram. Moreover, we prove that there is no pair of non-congruent planar polyominoes consisting of less than nine points that have equal discrete covariograms.

Let X and Y be separable Banach spaces and denote by 𝒮𝒮(X,Y ) the subset of ℒ(X,Y ) consisting of all strictly singular operators. We study various ordinal ranks on the set 𝒮𝒮(X,Y ). Our main results are summarized as follows. Firstly, we define a new rank r𝒮 on 𝒮𝒮(X,Y ). We show that r𝒮 is a co-analytic rank and that it dominates the rank ϱ introduced by Androulakis, Dodos, Sirotkin and Troitsky [Israel J. Math.169 (2009), 221–250]. Secondly, for every 1≤p<+∞, we construct a Banach space Yp with an unconditional basis such that 𝒮𝒮(ℓp,Yp) is a co-analytic non-Borel subset of ℒ(ℓp,Yp) yet every strictly singular operator T:ℓp→Yp satisfies ϱ(T)≤2. This answers a question of Argyros.

A line that intersects every member of a finite family F of convex sets in the plane is called a common transversal to F. In this paper we study some basic properties of T(k)-families: finite families of convex sets in the plane in which every subfamily of size at most k admits a common transversal. It is known that a T(k)-family admits a partial transversal of size α∣F∣ for some constant α(k) which is independent of F. Here it will be shown that (2/(k(k−1)))1/(k−2)≤α(k)≤((k−2)/(k−1)), which are the best bounds to date.

Gravity and surface-tension effects are examined for inviscid–inviscid interactions between two fluids close to a wall. The ratios of density and viscosity of the two fluids are taken to be small. A nonlinear integro–differential equation is found to govern the near-wall flow velocity, interface shape and pressure; analysis, computation and comparisons are then applied. Travelling-state solutions are of particular interest.

We apply a method of Davenport to improve several estimates for slim exceptional sets associated with the asymptotic formula in Waring’s problem. In particular, we show that the anticipated asymptotic formula in Waring’s problem for sums of seven cubes holds for all but O(N1/3+ε) of the natural numbers not exceeding N.

We prove algebraic transformations for the generating series of three Apéry-like sequences. As application, we provide new binomial representations for the sequences. We also illustrate a method that derives three new series for 1/π from a classical Ramanujan’s series.

We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at ±∞. We use this result to determine the asymptotic distribution of the eigenvalues.

We study the variance of the fluctuations in the number of lattice points in a ball and in a thin spherical shell of large radius centred at a Diophantine point.

We prove a stability version of the Prékopa–Leindler inequality.