Motivated by reformulating Furstenberg's $\times p,\times q$ conjecture via representations of a crossed product $C^*$-algebra, we show that in a discrete $C^*$-dynamical system $(A,\Gamma)$, the space of (ergodic) $\Gamma$-invariant states on $A$ is homeomorphic to a subspace of (pure) state space of $A\rtimes\Gamma$. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic $\Gamma$-invariant regular Borel probability measures on a compact Hausdorff space $X$ is equivalent to the classification a special type of irreducible representations of $C(X)\rtimes \Gamma$.

In our previous paper “The primes contain arbitrarily long polynomial progressions” [$Acta Math$., 201 (2008), 213–305] there were some minor errors in our definition of the polynomial forms and polynomial correlation conditions, which is unreasonably strong as written due to the overly loose bound on the degree of the polynomials involved. In this erratum we repair this issue by capping the degree of the polynomials more strongly, and adjusting some other relevant components of the argument accordingly.

We construct faithful actions of quantum permutation groups on connected compact metrizable spaces. This disproves a conjecture of Goswami.

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points <i>F</i>(<i>S</i>) of a nonexpansive mapping <i>S</i> and the set of solutions _<i>A</i> of the variational inequality for a monotone, Lipschitz continuous mapping <i>A</i>. We introduce a hybrid extragradient-like approximation method which is based on the well-known extragradient method and a hybrid (or outer approximation) method. The method produces three sequences which are shown to converge strongly to the same common element of {F(S)\cap\Omega_{A}}. As applications, the method provides an algorithm for finding the common fixed point of a nonexpansive mapping and a pseudocontractive mapping, or a common zero of a monotone Lipschitz continuous mapping and a maximal monotone mapping.

Assume that <i>F</i> is a nonlinear operator on a real Hilbert space <i>H</i> which is -strongly monotone and -Lipschitzian on a nonempty closed convex subset <i>C</i> of <i>H</i>. Assume also that <i>C</i> is the intersection of the fixed-point sets of a finite number of nonexpansive mappings on <i>H</i>. We construct an iterative algorithm with variable parameters which generates a sequence {<i>x</i> _<i>n</i>} from an arbitrary initial point <i>x</i> ₀ <i>H</i>. The sequence {<i>x</i> _<i>n</i>} is shown to converge in norm to the unique solution <i>u</i>  of the variational inequality