Tao and Vu showed that every centrally symmetric convex progression $C\subset \mathbb{Z}^d$ is contained in a generalized arithmetic progression of size $d^{O(d^2)} \# C$. Berg and Henk improved the size bound to $d^{O(d\log d)} \# C$. We obtain the bound $d^{O(d)} \# C$, which is sharp up to the implied constant and is of the same form as the bound in the continuous setting given by John’s theorem.

In 1945–1946, C. L. Siegel proved that an $n$-dimensional lattice $\unicode[STIX]{x1D6EC}$ of determinant $\text{det}(\unicode[STIX]{x1D6EC})$ has at most $m^{n^{2}}$ different sublattices of determinant $m\cdot \text{det}(\unicode[STIX]{x1D6EC})$. In 1997, the exact number of the different sublattices of index $m$ was determined by Baake. We present a systematic treatment for counting the sublattices and derive a formula for the number of the sublattice classes under unimodular equivalence.

Given complex numbers w1,…,wn, we define the weight w(X) of a set X of 0–1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors (x1,…,xn) in X. We present an algorithm which, for a set X defined by a system of homogeneous linear equations with at most r variables per equation and at most c equations per variable, computes w(X) within relative error ∊ > 0 in (rc)O(lnn-ln∊) time provided $|w_j| \leq \beta (r \sqrt{c})^{-1}$ for an absolute constant β > 0 and all j = 1,…,n. A similar algorithm is constructed for computing the weight of a linear code over ${\mathbb F}_p$. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.

This paper is concerned with the maximisation of the $k$-th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension $d$ as $k$ goes to infinity. We show that in any dimension maximisers exist for any given $k$, but that any sequence of maximisers degenerates as $k$ goes to infinity when the dimension is at most 10. Furthermore, we obtain specific upper and lower bounds for the injectivity radius of any sequence of maximisers. We also prove that flat Klein bottles maximising the $k$-th eigenvalue of the Laplacian exhibit the same behaviour. These results contrast with those obtained recently by Gittins and Larson, stating that sequences of optimal cuboids for either Dirichlet or Neumann boundary conditions converge to the cube no matter the dimension. We obtain these results via Weyl asymptotics with explicit control of the remainder in terms of the injectivity radius. We reduce the problem at hand to counting lattice points inside anisotropically expanding domains, where we generalise methods of Yu. Kordyukov and A. Yakovlev by considering domains that expand at different rates in various directions.

We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449–1466]. By well-known decompositions, it is sufficient to consider the case of affine cones $s+\mathfrak{c}$ , where $s$ is an arbitrary real vertex and $\mathfrak{c}$ is a rational polyhedral cone. For a given rational subspace  $L$ , we define the intermediate generating functions $S^{L}(s+\mathfrak{c})(\unicode[STIX]{x1D709})$ by integrating an exponential function over all lattice slices of the affine cone  $s+\mathfrak{c}$ parallel to the subspace  $L$ and summing up the integrals. We expose the bidegree structure in parameters $s$ and $\unicode[STIX]{x1D709}$ , which was implicitly used in the algorithms in our papers [Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra. Found. Comput. Math.12 (2012), 435–469] and [Intermediate sums on polyhedra: computation and real Ehrhart theory. Mathematika59 (2013), 1–22]. The bidegree structure is key to a new proof for the Baldoni–Berline–Vergne approximation theorem for discrete generating functions [Local Euler–Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of rational polytopes. Contemp. Math.452 (2008), 15–33], using the Fourier analysis with respect to the parameter  $s$ and a continuity argument. Our study also enables a forthcoming paper, in which we study intermediate sums over multi-parameter families of polytopes.

We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok in [Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449–1466]. For a given polytope 𝔭 with facets parallel to rational hyperplanes and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope 𝔭 parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step-polynomials. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory.

A classical theorem of Rogers states that for any convex body $K$ in $n$-dimensional Euclidean space there exists a covering of the space by translates of $K$ with density not exceeding $n\,\log \,n\,+\,n\,\log \,\log \,n\,+\,5$. Rogers’ theorem does not say anything about the structure of such a covering. We show that for sufficiently large values of $n$ the same bound can be attained by a covering which is the union of $O\left( \log \,n \right)$ translates of a lattice arrangement of $K$.

In this note we present inequalities relating the successive minima of an $o$-symmetric convex body and the successive inner and outer radii of the body. These inequalities join known inequalities involving only either the successive minima or the successive radii.

A Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice $\Gamma$ inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a continuous function and the uniformlattice Dirac comb, and its diffraction measure is periodic, with the dual lattice ${{\Gamma }^{*}}$ as lattice of periods. This statement remains true in the setting of a locally compact Abelian group whose topology has a countable base.

We prove polytopality of the generalized associahedra introduced in [5].

Lattices and $\mathbb{Z}$-modules in Euclidean space possess an infinitude of subsets that are images of the original set under similarity transformation. We classify such self-similar images according to their indices for certain $4D$ examples that are related to $4D$ root systems, both crystallographic and non-crystallographic. We encapsulate their statistics in terms of Dirichlet series generating functions and derive some of their asymptotic properties.

Partial answers are given to two questions. When does a lattice $\Lambda $ contain a sublattice ${\Lambda }'$ of index $N$ that is geometrically similar to $\Lambda $? When is the sublattice “clean”, in the sense that the boundaries of the Voronoi cells for ${\Lambda }'$ do not intersect $\Lambda $?

Let $M$ be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large $\lambda$ the number of lattice points in $\lambda M$ is given by $G\left( \lambda M \right)=V\left( \lambda M \right)+O\left( {{\lambda }^{d-1-\varepsilon \left( d \right)}} \right)$ for some positive $\varepsilon (d)$. Here we give for general convex bodies the weaker estimate

$$|G\left( \lambda M \right)-V\left( \lambda M \right)|\,\le \,\frac{1}{2}{{S}_{{{Z}^{d}}}}\left( M \right){{\lambda }^{d-1}}+o\left( {{\lambda }^{d-1}} \right)$$

where ${{S}_{{{Z}^{d}}}}\left( M \right)$ denotes the lattice surface area of $M$. The term ${{S}_{{{Z}^{d}}}}\left( M \right)$ is optimal for all convex bodies and $o\left( {{\lambda }^{d-1}} \right)$ cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of $M$.

Further we deal with families $\left\{ {{P}_{\lambda }} \right\}$ of convex bodies where the only condition is that the inradius tends to infinity. Here we have

$$|G\left( {{P}_{\lambda }} \right)-V\left( {{P}_{\lambda }} \right)|\,\le \,dV\left( {{P}_{\lambda }},\,K;1 \right)+o\left( S\left( {{P}_{\lambda }} \right) \right)$$

where the convex body $K$ satisfies some simple condition, $V\left( {{P}_{\lambda }},K;1 \right)$ is some mixed volume and $S\left( {{P}_{\lambda }} \right)$ is the surface area of ${{P}_{\lambda }}$.