Our main aim in this paper is to show that there is a partition of the reals into finitely many classes with ‘many’ forbidden distances, in the following sense: for each positive real x, there is a natural number n such that no two points in the same class are at distance $x/n$. In fact, more generally, given any infinite set $\{ c_n:n<\omega\}$ of positive rationals, there is a partition of the reals into three classes such that for each positive real x, there is some n such that no two points in the same class are at distance $c_nx$. This result is motivated by some questions in partition regularity.

We prove that the permutation representation of the finite orthogonal group $\Omega^{\varepsilon}(n,q)$, where $\varepsilon=+$ or $-$, on the set of anisotropic lines is multiplicity-free, if q is a power of 2 and $n\ge 6$ is even. This result is established by giving a description of orbitals of this action. The rank of this action is $(q^2+2q)/2$ if $\varepsilon=+$ and $n=6$, and $(q^2+2q+2)/2$ otherwise. Moreover, we compute the subdegrees of the orbitals of $\Omega^{\varepsilon}(n,q)$.

An RC-group is a unital group G with a distinguished compression base with respect to which G satisfies the Rickart projection and general comparison properties. We prove that a monotone $\sigma$-complete RC-group is a union of subgroups each of which is a lattice-ordered Dedekind $\sigma$-complete RC-group.

In a 2004 paper, Totaro asked whether a G-torsor X that has a zero-cycle of degree $d>0$ will necessarily have a closed étale point of degree dividing d, where G is a connected algebraic group. This question is closely related to several conjectures regarding exceptional algebraic groups. Totaro gave a positive answer to his question in the following cases: G simple, split, and of type $G_2$, type $F_4$, or simply connected of type $E_6$. We extend the list of cases where the answer is ‘yes’ to all groups of type $G_2$ and some nonsplit groups of type $F_4$ and $E_6$. No assumption on the characteristic of the base field is made. The key tool is a lemma regarding linkage of Pfister forms.

A certain sequence of weight $1/2$ modular forms arises in the theory of Borcherds products for modular forms for $\textrm{SL}_{2}(\Z)$. Zagier proved a family of identities between the coefficients of these weight $1/2$ forms and a similar sequence of weight $3/2$ modular forms, which interpolate traces of singular moduli. We obtain the analogous results for modular forms arising from Borcherds products for Hilbert modular forms.

For any given real number $\alpha$ with bounded partial quotients, we construct explicitly continuum many real numbers $\beta$ with bounded partial quotients for which the pair $(\alpha, \beta)$ satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.

The Skolem–Mahler–Lech theorem states that if $f(n)$ is a sequence given by a linear recurrence over a field of characteristic 0, then the set of m such that $f(m)$ is equal to 0 is the union of a finite number of arithmetic progressions in $m\ge 0$ and a finite set. We prove that if X is a subvariety of an affine variety Y over a field of characteristic 0 and q is a point in Y, and $\sigma$ is an automorphism of Y, then the set of m such that $\sigma^m({\bf q})$ lies in X is a union of a finite number of complete doubly-infinite arithmetic progressions and a finite set. We show that this is a generalisation of the Skolem–Mahler–Lech theorem.

Our first result is a ‘sum-product’ theorem for subsets A of the finite field ${{\mathbb F}_p}$, p prime, providing a lower bound on $\max (|A+A|, |A\cdot A|)$. The second and main result provides new bounds on exponential sums

\[\sum_{x_1,\dots,x_k\in A} \exp(2\pi ix_1\dotsc x_k\xi/p),\]

where $A\subset{{\mathbb F}_p}$.

We define a sequence of real functions which coincide with Li's coefficients at one and which allow us to extend Li's criterion for the Riemann Hypothesis to yield a necessary and sufficient condition for the existence of zero-free strips inside the critical strip $0<\Re(z)<1$. We study some of the properties of these functions, including their oscillatory behaviour.

Let $f\in \mathbb{Z}[ x,y]$ be a reducible homogeneous polynomial of degree 3. We show that $f(x,y)$ has an even number of prime factors as often as an odd number of prime factors.

For a variety X of dimension n in ${\mathbb P}^r,\ r\geq n(k+1)+k$, the kth secant order of X is the number $\mu_k(X)$ of $(k+1)$-secant k-spaces passing through a general point of the kth secant variety. We show that, if $r>n(k+1)+k$, then $\mu_k(X)=1$ unless X is k-weakly defective. Furthermore we give a complete classification of surfaces $X\subset{\mathbb P}^r,\ r>3k+2$, for which $\mu_k(X)>1$.

Associated to any complex simple Lie algebra is a non-reductive complex Lie algebra which we call the intermediate Lie algebra. We propose that these algebras can be included in both the magic square and the magic triangle to give an additional row and column. The extra row and column in the magic square correspond to the sextonions. This is a six-dimensional subalgebra of the split octonions which contains the split quaternions.

We show that for every natural number m a finitely generated metabelian group G embeds in a quotient of a metabelian group of type $\textit{FP}_m$. Furthermore, if $m \leq 4$, the group G can be embedded in a metabelian group of type $\textit{FP}_m$. For L a finitely generated metabelian Lie algebra over a field K and a natural number m we show that, provided the characteristic p of K is 0 or $p > m$, then L can be embedded in a metabelian Lie algebra of type $\textit{FP}_m$. This result is the best possible as for $0 < p\leq m$ every metabelian Lie algebra over K of type $\textit{FP}_m$ is finite dimensional as a vector space.

We investigate the problem of finding a set of prime divisors of the order of a finite group, such that no two irreducible characters are in the same p-block for all primes p in the set. Our main focus is on the simple and quasi-simple groups. For results on the alternating and symmetric groups and their double covers, some combinatorial results on the cores of partitions are proved, which may be of independent interest. We also study the problem for groups of Lie type. The sporadic groups (and their relatives) are checked using GAP.

We say that a bounded linear operator T acting on a Banach space B is antisupercyclic if for any $x\in B$ either $T^nx=0$ for some positive integer n or the sequence $\{T^nx/\|T^nx\|\}$ weakly converges to zero in B. Antisupercyclicity of T means that the angle criterion of supercyclicity is not satisfied for T in the strongest possible way. Normal antisupercyclic operators and antisupercyclic bilateral weighted shifts are characterized.

As for the Volterra operator V, it is proved that if $1\leq p\leq\infty$ and any $f\in L_p[0,1]$ then the limit $\lim_{n\to\infty} (n!\|V^nf\|_p)^{1/n}$ does exist and equals $1-\inf\,\hbox{\rm supp}\,(f)$. Upon using this asymptotic formula it is proved that the operator V acting on the Banach space $L_p[0,1]$ is antisupercyclic for any $p\in(1,\infty)$. The same statement for $p=1$ or $p=\infty$ is false. The analogous results are proved for operators $V^zf(x)=(1/{\Gamma(z)})\int_0^x f(t) (x-t)^{z-1}\,dt$ when the real part of $z\in{\mathbb C}$ is positive.

This paper is devoted to the long-time behavior of solutions to the Cauchy problem of the porous medium equation $u_t=\Delta(u^m)-u^p$ in $\mathbb{R}^n\times (0,\infty)$ with $(1-2/n)_{+}<m<1$ and the critical exponent $p=m+2/n$. For the strictly positive initial data $u(x,0)=O(1+|x|)^{-k}$ with $n+mn(2-n+nm)/(2[2-m+mn(1-m)])\leq k<2/(1-m)$, we prove that the solution of the above Cauchy problem converges to a fundamental solution of $u_t=\Delta(u^m)$ with an additional logarithmic anomalous decay exponent in time as $t\to \infty$.