Given a set D of positive real numbers, let Xn(D) denote the graph with ℝn as the vertex set such that two points are joined if their distance is in D. Bukh conjectured in [Measurable sets with excluded distances. Geom. Funct. Anal.18 (2008), 668–697] that if D is algebraically independent, then Chr(Xn(D)), the chromatic number of Xn(D), is finite. Here we prove that Chr(Xn(D)) is countable and that, if n=2 , even the coloring number is countable. Furthermore, we prove that Chr (Y ) is countable, where Y is the following graph on ℂn: let 𝔽 be a countable subfield of ℂ and let D⊆ℂ be algebraically independent over 𝔽; join a,b∈ℂn if there is some p(x,y)∈𝔽[x,y] such that p(x,x) is identically zero and p(a,b)≠0 is algebraic over some d∈𝔽∪D.

We develop Weyl differencing and Hua-type lemmata for a class of multidimensional exponential sums. We then apply our estimates to bound the number of variables required to establish an asymptotic formula for the number of solutions of a system of diophantine equations arising from the study of linear spaces on hypersurfaces. For small values of the degree and dimension, our results are superior to those stemming from the author’s earlier work on Vinogradov’s mean value theorem.

For a primitive root g modulo a prime p≥1 we obtain upper bounds on the gaps between the residues modulo p of the N consecutive powers agn, n=1,…,N, which is uniform over all integers a with gcd (a,p)=1.

Paucity is established for a system of diagonal diophantine equations, in which the degrees are the odd numbers in ascending order.

We consider exponential sums with x-coordinates of points qG and q−1G where G is a point of order T on an elliptic curve modulo a prime p and q runs through all primes up to N (with gcd (q,T)=1 in the case of the points q−1G). We obtain a new bound on exponential sums with q−1G and correct an imprecision in the work of W. D. Banks, J. B. Friedlander, M. Z. Garaev and I. E. Shparlinski on exponential sums with qG. We also note that similar sums with g1/q for an integer g with gcd (g,p)=1 have been estimated by J. Bourgain and I. E. Shparlinski.

We study the distribution of the size of Selmer groups and Tate–Shafarevich groups arising from a 2-isogeny and its dual 2-isogeny for elliptic curves En:y2=x3−n3. We show that the 2-ranks of these groups all follow the same distribution. The result also implies that the mean value of the 2-rank of the corresponding Tate–Shafarevich groups for square-free positive integers n≤X is as X→∞. This is quite different from quadratic twists of elliptic curves with full 2-torsion points over ℚ [M. Xiong and A. Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math. 219 (2008), 523–553], where one Tate–Shafarevich group is almost always trivial while the other is much larger.

Let a be an integer different from 0, ±1, or a perfect square. We consider a conjecture of Erdős which states that #{p:ℓa(p)=r}≪εrε for any ε>0, where ℓa(p) is the order of a modulo p. In particular, we see what this conjecture says about Artin’s primitive root conjecture and compare it to the generalized Riemann hypothesis and the ABC conjecture. We also extend work of Goldfeld related to divisors of p+a and the order of a modulo p.

Let G be a subgroup of the symmetric group Sn, and let δG=∣Sn/G∣−1 where ∣Sn/G∣ is the index of G in Sn. Then there are at most On,ϵ(Hn−1+δG+ϵ) monic integer polynomials of degree n that have Galois group G and height not exceeding H, so there are only a “few” polynomials having a “small” Galois group.

For a probability measure μ on a real separable Hilbert space H, we are interested in “volume-based” approximations of the d-dimensional least squares error of μ, i.e., least squares error with respect to a best fit d-dimensional affine subspace. Such approximations are given by averaging real-valued multivariate functions which are typically scalings of squared (d+1)-volumes of (d+1)-simplices in H. Specifically, we show that such averages are comparable to the square of the d-dimensional least squares error of μ, where the comparison depends on a simple quantitative geometric property of μ. This result is a higher dimensional generalization of the elementary fact that the double integral of the squared distances between points is proportional to the variance of μ. We relate our work to two recent algorithms, one for clustering affine subspaces and the other for Monte-Carlo singular value decomposition based on volume sampling.

Extending classical results of Nair and Tenenbaum, we provide general, sharp upper bounds for sums of the type where x,y,u,v have comparable logarithms, F belongs to a class defined by a weak form of sub-multiplicativity, and the Qj are arbitrary binary forms. A specific feature of the results is that the bounds are uniform within the F-class and that, as in a recent version given by Henriot, the dependency with respect to the coefficients of the Qj is made explicit. These estimates play a crucial rôle in the proof, published separately by the authors, of Manin’s conjecture for Châtelet surfaces.

We prove the following generalization of the ham sandwich theorem, conjectured by Imre Bárány. Given a positive integer k and d nice measures μ1,μ2,…,μd in ℝd such that μi(ℝd)=k for all i, there is a partition of ℝd into k interior-disjoint convex parts C1,C2,…,Ck such that μi (Cj)=1 for all i,j. If k=2 , this gives the ham sandwich theorem. This result was proved independently by R. N. Karasev.

Let T be an algebraic torus over ℚ such that T(ℝ) is compact. Assuming the generalized Riemann hypothesis, we give a lower bound for the size of the class group of T modulo its n-torsion in terms of a small power of the discriminant of the splitting field of T. As a corollary, we obtain an upper bound on the n-torsion in that class group. This generalizes known results on the structure of class groups of complex multiplication fields.

The paper extends the fundamental existence assertion for probability contents and measures with given marginals: the extension is from algebras to lattices, and thus is in accord with an actual trend in measure and integration. The proof of the basic theorem is a rapid application of a former Hahn–Banach type separation theorem.

Consider a random simplex in a d-dimensional convex body which is the convex hull of d+1 random points from the body. We study the following question: as a function of the convex body, is the expected volume of such a random simplex monotone non-decreasing under inclusion? We show that this is true when d is 1 or 2, but does not hold for d≥4. We also prove similar results for higher moments of the volume of a random simplex, in particular for the second moment, which corresponds to the determinant of the covariance matrix of the convex body. These questions are motivated by the slicing conjecture.

In the so-called Bernoulli model of the kinetic theory of gases, where (1) the particles are dimensionless points, (2) they are contained in a cube container, (3) no attractive or exterior force is acting on them, (4) there is no collision between the particles, (5) the collisions against the walls of the container are according to the law of elastic collision, we deduce from Newtonian mechanics two basic probabilistic limit laws about particle-counting. The first one is a global central limit theorem, and the second one is a local Poisson law. The key fact is that we can prove the “realistic limit”: first the time interval is fixed and the number of particles tends to infinity (where the density particle/volume remains a fixed constant), and then, in the second step, the time tends to infinity.

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain Ω in ℝN. We consider deformations ϕ(Ω) of Ω obtained by means of a locally Lipschitz homeomorphism ϕ and we estimate the variation of the eigenfunctions and eigenvalues upon variation of ϕ. We prove general stability estimates without assuming uniform upper bounds for the gradients of the maps ϕ. As an application, we obtain estimates on the rate of convergence for eigenvalues and eigenfunctions when a domain with an outward cusp is approximated by a sequence of Lipschitz domains.

We suggest an alternative mathematical model for the electron in dimension 1+2. We think of our (1+2)-dimensional spacetime as an elastic continuum whose material points can experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose a coframe and a density. We then add an extra (third) spatial dimension, extend our coframe and density into dimension 1+3, choose a conformally invariant Lagrangian proportional to axial torsion squared, roll up the extra dimension into a circle so as to incorporate mass and return to our original (1+2)-dimensional spacetime by separating out the extra coordinate. The main result of our paper is the theorem stating that our model is equivalent to the Dirac equation in dimension 1+2. In the process of analysing our model we also establish an abstract result, identifying a class of nonlinear second order partial differential equations which reduce to pairs of linear first order equations.

For q>1, the set Fq of real numbers which can be expanded in base q with respect to the digit set {0,1,q} is just a self-similar set with overlaps. We consider the subset of Fq whose elements have a unique expansion and calculate its Hausdorff dimension for the case where .