It is shown that if $n>2k+4$ and if $A\subseteq {\mathbb{Z}}^n$ is a set of upper density $ϵ>0$, then—in a sense depending on $ϵ$—all large dilates of any given k-dimensional simplex $△=\{0,v_1,\dots ,v_k\}\subset {\mathbb{Z}}^n$ can be embedded in $A$. A simplex $△$ can be embedded in the set $A$ if $A$ contains simplex ${△}^{\prime }$, which is isometric to $△$. Moreover, the same is true if only $△\subset {\mathbb{R}}^n$ is assumed, and $△$ satisfies some immediate necessary conditions.

The proof uses techniques of harmonic analysis developed for the continuous case, as well as a variant of the circle method due to Siegel [S]

We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, using a generalization of the Wirtinger inequality for the comass. Using a model for the classifying space ${\mathrm{BS}}^3$ built inductively out of ${\mathrm{BS}}^1$, we prove that the symmetric metrics of certain two-point homogeneous manifolds turn out not to be the systolically optimal metrics on those manifolds. We point out the unexpected role played by the exceptional Lie algebra $E_7$ in systolic geometry, via the calculation of Wirtinger constants. Using a technique of pullback with controlled systolic ratio, we calculate the optimal systolic ratio of the quaternionic projective plane, modulo the existence of a Joyce manifold with Spin($7$)-holonomy and unit middle-dimensional Betti number

We define Floer homology for a time-independent or autonomous Hamiltonian on a symplectic manifold with contact-type boundary under the assumption that its $1$-periodic orbits are transversally nondegenerate. Our construction is based on Morse-Bott techniques for Floer trajectories. Our main motivation is to understand the relationship between the linearized contact homology of a fillable contact manifold and the symplectic homology of its filling