A Diophantine m-tuple is a set A of m positive integers such that ab+1 is a perfect square for every pair a,b of distinct elements of A. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by x. In doing so, we extend two existing tools in ways that might be of independent interest. The Erdős–Turán inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also adapt an argument of Hooley on the equidistribution of solutions of polynomial congruences to handle reducible quadratic polynomials.

An interval in a combinatorial structure R is a set I of points that are related to every point in R∖I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes—this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: an arbitrary structure S of size n belonging to a class 𝒞 can be embedded into a simple structure from 𝒞 by adding at most f(n) elements. We prove such results when 𝒞 is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than two. The functions f(n) in these cases are 2, ⌈log 2(n+1)⌉, ⌈(n+1)/2⌉, ⌈(n+1)/2⌉, ⌈log 4(n+1)⌉, ⌈log 3(n+1)⌉ and 1, respectively. In each case these bounds are the best possible.

We give improved bounds for our theorem in [W. T. Gowers and J. Wolf, The true complexity of a system of linear equations. Proc. London Math. Soc. (3) 100 (2010), 155–176], which shows that a system of linear forms on 𝔽np with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of 𝔽np. While in [W. T. Gowers and J. Wolf, The true complexity of a system of linear equations. Proc. London Math. Soc. (3) 100 (2010), 155–176] the dependence between the uniformity of the set and the resulting error in the average over the linear system was of tower type, we now obtain a doubly exponential relation between the two parameters. Instead of the structure theorem for bounded functions due to Green and Tao [An inverse theorem for the Gowers U3(G) norm. Proc. Edinb. Math. Soc. (2) 51 (2008), 73–153], we use the Hahn–Banach theorem to decompose the function into a quadratically structured plus a quadratically uniform part. This new decomposition makes more efficient use of the U3 inverse theorem [B. J. Green and T. Tao, An inverse theorem for the Gowers U3(G) norm. Proc. Edinb. Math. Soc. (2) 51 (2008), 73–153].

We give a short and “soft” proof of the asymptotic orthogonality of Fourier coefficients of Poincaré series for classical modular forms as well as for Siegel cusp forms, in a qualitative form.

In this paper, we study how close the terms of a finite arithmetic progression can get to a perfect square. The answer depends on the initial term, the common difference and the number of terms in the arithmetic progression.

Let 𝒟=(dn)∞n=1 be a sequence of integers with dn≥2, and let (i,j) be a pair of strictly positive numbers with i+j=1. We prove that the set of x∈ℝ for which there exists some constant c(x)≧0 such that is one-quarter winning (in the sense of Schmidt games). Thus the intersection of any countable number of such sets is of full dimension. This, in turn, establishes the natural analogue of Schmidt’s conjecture within the framework of the de Mathan–Teulié conjecture, also known as the “mixed Littlewood conjecture”.

We show that the conjecture of Kannan, Lovász, and Simonovits on isoperimetric properties of convex bodies and log-concave measures is true for log-concave measures of the form ρ(∣x∣B) dx on ℝn and ρ(t,∣x∣B) dx on ℝ1+n, where ∣x∣B is the norm associated to any convex body B already satisfying the conjecture. In particular, the conjecture holds for convex bodies of revolution.

Starting with a paper of Jacobson from the 1960s, many authors became interested in characterizing all algebraic number fields in which each integer is the sum of pairwise distinct units. Although there exist many partial results for number fields of low degree, a full characterization of these number fields is still not available. Narkiewicz and Jarden posed an analogous question for sums of units that are not necessarily distinct. In this paper we propose a generalization of these problems. In particular, for a given rational integer n we consider the following problem. Characterize all number fields for which every integer is a linear combination of finitely many units εi in a way that the coefficients ai∈ℕ are bounded by n. The paper gives several partial results on this problem. In our proofs we exploit the fact that these representations are related to symmetric beta expansions with respect to Pisot bases.

We describe some aspects of spectral theory that involve algebraic considerations but need no analysis. Some of the important applications of the results are to the algebra of n×n matrices with entries that are polynomials or more general analytic functions.

We prove that an algebraic subvariety of a Shimura variety is weakly special if and only if analytic components of its preimage in the symmetric space are algebraic. We also prove an analogous result in the case of abelian varieties.

We prove that if 3|d, then the d-dimensional balls are m-divisible for every m large enough. In particular, the three-dimensional balls are m-divisible for every m≥22.

An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. These highly symmetric convex bodies lie at the crossroads of several fields, including convex geometry, algebraic geometry, and optimization. We present a self-contained theory of orbitopes, with particular emphasis on instances arising from the groups SO(n) and O(n); these include Schur–Horn orbitopes, tautological orbitopes, Carathéodory orbitopes, Veronese orbitopes, and Grassmann orbitopes. We study their face lattices, algebraic boundaries, and representations as spectrahedra or projected spectrahedra.

The primary aim of this paper is to determine how two convex bodies are related if their respective normal bundles are equal or, in a certain sense, close to each other. In connection with this problem some pertinent results concerning parallel bodies and the Steiner point are also discussed.

We consider uniformly elliptic, second-order, linear partial differential equations depending on three variables in bounded domains. We obtain interior Hölder estimates for the first derivatives of the bounded solutions independent of the regularity assumptions of the differential operator.

It is proved that the simplex is a strict local minimum for the volume product, 𝒫(K)=min z∈int(K)|K||Kz|, in the Banach–Mazur space of n-dimensional (classes of) convex bodies. Linear local stability in the neighborhood of the simplex is proved as well. The proof consists of an extension to the non-symmetric setting of methods that were recently introduced by Nazarov, Petrov, Ryabogin and Zvavitch, as well as proving results of independent interest concerning stability of square order of volumes of polars of non-symmetric convex bodies.

Let P=A×A⊂𝔽p×𝔽p, p a prime. Assume that P=A×A has n elements, n<p. See P as a set of points in the plane over 𝔽p. We show that the pairs of points in P determine lines, where c is an absolute constant. We derive from this an incidence theorem: the number of incidences between a set of n points and a set of n lines in the projective plane over 𝔽p (n<p)is bounded by , where C is an absolute constant.

Schneider posed the problem of determining the maximal value of the affine invariant ∣ΠK∣/∣K∣d−1, where ΠK is the projection body of the d-dimensional convex body K. Some three-dimensional conjectures of Brannen, related to Schneider’s problem, are confirmed. Namely, we determine the maximal value of ∣ΠK∣/∣K∣2 in the class of three-dimensional zonoids, cones and double cones. Equality cases are, also, investigated. Moreover, results related to a conjecture of Petty, concerning the minimal value of the above quantity, are obtained. In particular, we provide a negative answer to a question of Martini and Mustafaev.

The time-dependent motion of water waves with a parametrically defined free surface is mapped to a fixed time-independent rectangle by an arbitrary transformation. The emphasis is on the general properties of transformations. Special cases are algebraic transformations based on transfinite interpolation, conformal mappings, and transformations generated by nonlinear elliptic partial differential equations. The aim is to study the effect of transformation on variational principles for water waves such as Luke’s Lagrangian formulation, Zakharov’s Hamiltonian formulation, and the Benjamin–Olver Hamiltonian formulation. Several novel features are exposed using this approach: a conservation law for the Jacobian, an explicit form for surface re-parameterization, inner versus outer variations and their role in the generation of hidden conservation laws of the Laplacian. Also some of the differential geometry of water waves becomes explicit. The paper is restricted to the case of planar motion, with a preliminary discussion of the extension to three-dimensional water waves.

We characterize straightness of digital curves in the integer plane by means of difference operators. Earlier definitions of digital rectilinear segments have used, respectively, Rosenfeld’s chord property, word combinatorics, Reveillès’ double Diophantine inequalities, and the author’s refined hyperplanes. We prove that all these definitions are equivalent. We also characterize convexity of integer-valued functions on the integers with the help of difference operators.