Thanks to Szemerédi's theorem on sets with no long arithmetic progressions, an elementary trick is used here to show that for a given positive integer $h$ and a given set $U$ of residue classes modulo $n$ with positive density, there exists a dense subset $V$ of $U$; that is, $U\smallsetminus V$ is very small, such that the sumset $hV$ is included in the restricted sumset $h\times U$. The next step is to obtain information on the structure of $V$ from Kneser's theorem on the sum of sets in an abelian group, and to use this for studying the structure of $U$ itself. Finally, this idea is used in the paper to derive some new values of a function related to the Erdős–Ginzburg–Ziv theorem.

A new lower bound on $\max \{|A+A|,|A\cdot A|\}$ is given, where $A$ is a finite set of complex numbers.

A lower and an upper bound are proved for the maximum size of a subset of $\mathbb{Z}/p\mathbb{Z}$ containing no solutions of a fixed linear equation.

It is proved that, if $P$ is a polynomial with integer coefficients, having degree 2, and $1>\varepsilon>0$, then $n(n-1)\cdots(n-k+1)=P(m)$ has only finitely many natural solutions $(m,n,k)$, $n\ge k>n\varepsilon$, provided that the $abc$ conjecture is assumed to hold under Szpiro's formulation.

As an application of the vector sieve and uniform estimates on the Fourier coefficients of cusp forms of half-integral weight, it is shown that any sufficiently large number $n\equiv 3$ (mod 24) with $5 \nmid n$ is expressible as a sum of three squares of integers having at most 521 prime factors.

It is proved that each mod 2 homology class of a compact Nash manifold can be represented by a semialgebraic arc-symmetric subset.

A proof is given that a permutation group in which different finite sets have different stabilizers cannot satisfy any group law. For locally compact topological groups with this property, almost all finite subsets of the group are shown to generate free subgroups. Consequences of these theorems are derived for: Thompson's group $F$, weakly branch groups, automorphism groups of regular trees, and profinite groups with alternating composition factors of unbounded degree.

The famous Burnside–Schur theorem states that every primitive finite permutation group containing a regular cyclic subgroup is either 2-transitive or isomorphic to a subgroup of a 1-dimensional affine group of prime degree. It is known that this theorem can be expressed as a statement on Schur rings over a finite cyclic group. Generalizing the latter, Schur rings are introduced over a finite commutative ring, and an analogue of this statement is proved for them. Also, the finite local commutative rings are characterized in permutation group terms.

A group $G$ is said to have periodic cohomology with period $q$ after $k$ steps, if the functors $H^{i}(G,-)$ and $H^{i+q}(G,-)$ are naturally equivalent for all $i>k$. Mislin and the author have conjectured that periodicity in cohomology after some steps is the algebraic characterization of those groups $G$ that admit a finite-dimensional, free $G$-CW-complex, homotopy equivalent to a sphere. This conjecture was proved by Adem and Smith under the extra hypothesis that the periodicity isomorphisms are given by the cup product with an element in $H^q (G,\mathbb{Z})$. It is expected that the periodicity isomorphisms will always be given by the cup product with an element in $H^q (G,\mathbb{Z})$; this paper shows that this is the case if and only if the group $G$ admits a complete resolution and its complete cohomology is calculated via complete resolutions. It is also shown that having the periodicity isomorphisms given by the cup product with an element in $H^q (G,\mathbb{Z})$ is equivalent to silp $G$ being finite, where silp $G$ is the supremum of the injective lengths of the projective $\mathbb{Z}G$-modules.

A new inequality concerning generalized characters of $p$-groups is obtained and applications to bounding the number of irreducible characters in blocks of finite groups are given.

The existence of periodic solutions for the nonlinear asymmetric oscillator $ x''{+}\alpha x^+{-}\beta x^- {=} h(t),\qquad (' = d/dt) $ is discussed, where $\alpha, \beta$ are positive constants satisfying $ {1}/{\sqrt{\alpha}} + {1}/{\sqrt{\beta}} = {2}/{n} $ for some positive integer $n\in {\bf \mathbb{N}}$ and $h(t)\in L^\infty(0,2\pi)$ is $2\pi$-periodic with $x^{\pm}=\max\{\pm x, 0\}$.

Effective formulae are found for the complex geodesics in the symmetrized bidisc.

The authors of this paper study positive supersolutions to the elliptic equation $-\Delta U=c|x|^{-s}u^p$ in Cone-like domains of $\mathbb{R}^N$ ($N\ge 2$), where $p,s\in\mathbb{R}$ and $c>0$. They prove that in the sublinear case $p<1$ there exists a critical exponent $p_\ast<1$ such that the equation has a positive supersolution if and only if $-\infty<p<p_\ast$. The value of $p_\ast$ is determined explicitly by $s$ and the geometry of the cone.

Using Morse theory and the truncation technique, a proof is given of the existence of at least three nontrivial solutions for a class of $p$-Laplacian equations. When $p=2$, the existence of four nontrivial solutions is also considered.

The relationship between two natural definitions of the relative pressure function, for a locally constant potential function and a factor map from a shift of finite type, is clarified by showing that they coincide almost everywhere with respect to every invariant measure. With a suitable extension of one of the definitions, the same holds true for any continuous potential function.

Thomas conjectured that there is no such thing as an almost prime element in a commutative radical Banach algebra. This paper shows that there is such an element. However, it still leaves open the question of whether there is an element that is almost prime, but not prime.

For any nonnegative self-adjoint operators $A$ and $B$ in a separable Hilbert space, the Trotter-type formula $[;(e^{i2tA/n}+e^{i2tB/n})/2]^n$ is shown to converge strongly in the norm closure of $\dom(A^{1/2})\cap\dom(B^{1/2})$ for some subsequence and for almost every $t\in\mathbb{R}$. This result extends to the degenerate case, and to Kato-functions following the method of T. Kato (see ‘Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroup’, Topics in functional analysis (ed. M. Kac, Academic Press, New York, 1978) 185–195). Moreover, the restrictions on the convergence can be removed by considering functions other than the exponential.

In his thesis (Mem. Amer. Math. Soc. 42 (1962)) A. Liulevicius defined Steenrod squaring operations $Sq^k$ on the cohomology ring of any cocommutative Hopf algebra over $Z/2$. Later, J. P. May showed that these operations satisfy Adem relations, interpreted so that $Sq^0$ is not the unit but an independent operation. Thus, these Adem relations are homogeneous of length two in the generators. This paper is concerned with the bigraded algebra $\cal {B}$ that is generated by elements $Sq^k$ and subject to Adem relations; it shows that the Cartan formula gives a well-defined coproduct on $\cal {B}$. Also, it is shown that $\cal {B}$ with both multiplication and comultiplication should be considered neither a Hopf algebra nor a bialgebra, but another kind of structure, for which the name ‘algebra with coproducts’ is proposed in the paper.