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Let $K$ be a number field of degree $d$. Then every ideal $I$ in the ring of integers ${\mathcal O}_K$ contains infinitely many primitive elements, i.e. elements of degree $d$. A bound on smallest height of such an element in $I$ follows from some recent developments in the direction of a 1998 conjecture of W. Ruppert. We prove a very explicit bound like this in the case of quadratic fields. Further, we consider primitive elements in an ideal outside of a finite union of other ideals and prove a bound on the height of a smallest such element. Our main tool is a result on points of small norm in a lattice outside of an algebraic hypersurface and a finite union of sublattices of finite index, which we prove by blending two previous Diophantine avoidance results. We also obtain an avoidance result like this for lattice points in the positive orthant in $\mathbb{R}^d$ and use it to obtain a small-height totally positive primitive element in an ideal of a totally real number field outside of a finite union of other ideals. Additionally, we use our avoidance method to prove a bound on the Mahler measure of a generating non-sparse polynomial for a given number field. Finally, we produce a bound on the height of a smallest primitive generator for a principal ideal in a quadratic number field., Comment: 16 pages

Given a lattice $L$ in the plane, we define the affiliated deep hole lattice $H(L)$ to be spanned by a shortest vector of $L$ and a deep hole of $L$ contained in the triangle with sides corresponding to the shortest basis vectors. We study the geometric and arithmetic properties of deep hole lattices. In particular we investigate conditions on $L$ under which $H(L)$ is well-rounded and prove that $H(L)$ is defined over the same field as $L$. For the period lattice corresponding to an isomorphism class of elliptic curves, we produce a finite sequence of deep hole lattices ending with a well-rounded lattice which corresponds to a point on the boundary arc of the fundamental strip under the action of $\operatorname{SL}_2(\mathbb{Z})$ on the upper halfplane. In the case of CM elliptic curves, we prove that all elliptic curves generated by this sequence are isogenous to each other and produce bounds on the degree of isogeny. Finally, we produce a counting estimate for the planar lattices with a prescribed deep hole lattice., Comment: 12 pages, 2 figures; to appear in Research in Number Theory

We establish a new version of Siegel's lemma over a number field $k$, providing a bound on the maximum of heights of basis vectors of a subspace of $k^N$, $N \geq 2$. In addition to the small-height property, the basis vectors we obtain satisfy certain sparsity condition. Further, we produce a nontrivial bound on the heights of all the possible subspaces generated by subcollections of these basis vectors. Our bounds are absolute in the sense that they do not depend on the field of definition. The main novelty of our method is that it uses only linear algebra and does not rely on the geometry of numbers or the Dirichlet box principle employed in the previous works on this subject., Comment: 16 pages; to appear in Research in the Mathematical Sciences

In this editorial survey we introduce the special issue of the journal Communications in Mathematics on the topic in the title of the article. Our main goal is to briefly outline some of the main aspects of this important area at the intersection of theory and applications, providing the context for the articles showcased in this special issue., Comment: Communications in Mathematics 31 (2023), no. 2, 251--263

A lattice $\Lambda$ is said to be an extension of a sublattice $L$ of smaller rank if $L$ is equal to the intersection of $\Lambda$ with the subspace spanned by $L$. The goal of this paper is to initiate a systematic study of the geometry of lattice extensions. We start by proving the existence of a small-determinant extension of a given lattice, and then look at successive minima and covering radius. To this end, we investigate extensions (within an ambient lattice) preserving the successive minima of the given lattice, as well as extensions preserving the covering radius. We also exhibit some interesting arithmetic properties of deep holes of planar lattices., Comment: 18 pages, 4 figures; to appear in Monatshefte f\"ur Mathematik

We consider multivariable polynomials over a fixed number field, linear in some of the variables. For a system of such polynomials satisfying certain technical conditions we prove the existence of search bounds for simultaneous zeros with respect to height. For a single such polynomial, we prove the existence of search bounds with respect to height for zeros lying outside of a prescribed algebraic set. We also obtain search bounds in the case of homogeneous multilinear polynomials, which are related to a so-called "sparse" version of Siegel's lemma. Among the tools we develop are height inequalities that are of some independent interest., Comment: 15 pages; to appear in Journal of Number Theory

We focus on two important classes of lattices, the well-rounded and the cyclic. We show that every well-rounded lattice in the plane is similar to a cyclic lattice, and use this cyclic parameterization to count planar well-rounded similarity classes defined over a fixed number field with respect to height. We then investigate cyclic properties of the irreducible root lattices in arbitrary dimensions, in particular classifying those that are simple cyclic, i.e. generated by rotation shifts of a single vector. Finally, we classify cyclic, simple cyclic and well-rounded cyclic lattices coming from rings of integers of Galois algebraic number fields., Comment: 17 pages; to appear in Moscow Journal of Combinatorics and Number Theory

Let $L$ be a full-rank lattice in $\mathbb R^d$ and write $L^+$ for the semigroup of all vectors with nonnegative coordinates in $L$. We call a basis $X$ for $L$ positive if it is contained in $L^+$. There are infinitely many such bases, and each of them spans a conical semigroup $S(X)$ consisting of all nonnegative integer linear combinations of the vectors of $X$. Such $S(X)$ is a sub-semigroup of $L^+$, and we investigate the distribution of the gaps of $S(X)$ in $L^+$, i.e. the points in $L^+ \setminus S(X)$. We describe some basic properties and counting estimates for these gaps. Our main focus is on the restrictive successive minima of $L^+$ and of $L^+ \setminus S(X)$, for which we produce bounds in the spirit of Minkowski's successive minima theorem and its recent generalizations. We apply these results to obtain analogous bounds for the successive minima with respect to Weil height of totally positive sub-semigroups of ideals in totally real number fields., Comment: 14 pages, to appear in Advances in Geometry

We introduce maximal and average coherence on lattices by analogy with these notions on frames in Euclidean spaces. Lattices with low coherence can be of interest in signal processing, whereas lattices with high orthogonality defect are of interest in sphere packing problems. As such, coherence and orthogonality defect are different measures of the extent to which a lattice fails to be orthogonal, and maximizing their quotient (normalized for the number of minimal vectors with respect to dimension) gives lattices with particularly good optimization properties. While orthogonality defect is a fairly classical and well-studied notion on various families of lattices, coherence is not. We investigate coherence properties of a nice family of algebraic lattices coming from rings of integers in cyclotomic number fields, proving a simple formula for their average coherence. We look at some examples of such lattices and compare their coherence properties to those of the standard root lattices., Comment: Communications in Mathematics 31 (2023), no. 1, 57-72

Given a primitive collection of vectors in the integer lattice, we count the number of ways it can be extended to a basis by vectors with sup-norm bounded by $T$, producing an asymptotic estimate as $T \to \infty$. This problem can be interpreted in terms of unimodular matrices, as well as a representation problem for a class of multilinear forms. In the $2$-dimensional case, this problem is also connected to the distribution of Farey fractions. As an auxiliary lemma we prove a counting estimate for the number of integer lattice points of bounded sup-norm in a hyperplane in~$\mathbb R^n$. Our main result on counting basis extensions also generalizes to arbitrary lattices in~$\mathbb R^n$. Finally, we establish some basic properties of sparse representations of integers by multilinear forms., Comment: 15 pages; to appear in the Proceedings of AMS

Let $L$ be a lattice of full rank in $n$-dimensional real space. A vector in $L$ is called $i$-sparse if it has no more than $i$ nonzero coordinates. We define the $i$-th successive sparsity level of $L$, $s_i(L)$, to be the minimal $s$ so that $L$ has $s$ linearly independent $i$-sparse vectors, then $s_i(L) \leq n$ for each $1 \leq i \leq n$. We investigate sufficient conditions for $s_i(L)$ to be smaller than $n$ and obtain explicit bounds on the sup-norms of the corresponding linearly independent sparse vectors in~$L$. This result can be viewed as a partial sparse analogue of Minkowski's successive minima theorem. We then use this result to study virtually rectangular lattices, establishing conditions for the lattice to be virtually rectangular and determining the index of a rectangular sublattice. We further investigate the $2$-dimensional situation, showing that virtually rectangular lattices in the plane correspond to elliptic curves isogenous to those with real $j$-invariant. We also identify planar virtually rectangular lattices in terms of a natural rationality condition of the geodesics on the modular curve carrying the corresponding points., Comment: 18 pages; to appear in Research in the Mathematical Sciences

Let $F(\boldsymbol x)$ be a homogeneous polynomial in $n \ge 1$ variables of degree $1 \leq d \leq n$ with integer coefficients so that its degree in every variable is equal to $1$. We give some sufficient conditions on $F$ to ensure that for every integer $b$ there exists an integer vector $\boldsymbol a$ such that $F(\boldsymbol a) = b$. The conditions provided also guarantee that the vector $\boldsymbol a$ can be found in a finite number of steps., Comment: 9 pages, to appear in Research in Number Theory

Nearly orthogonal lattices were formally defined in [4], where their applications to image compression were also discussed. The idea of ``near orthogonality" in $2$-dimensions goes back to the work of Gauss. In this paper, we focus on well-rounded nearly orthogonal lattices in~$\mathbb R^n$ and investigate their geometric and optimization properties. Specifically, we prove that the sphere packing density function on the space of well-rounded lattices in dimension $n\geq 3$ does not have any local maxima on the nearly orthogonal set and has only one local minimum there: at the integer lattice~$\mathbb Z^n$. Further, we show that the nearly orthogonal set cannot contain any perfect lattices for~$n \geq 3$, although it contains multiple eutactic (and even strongly eutactic) lattices in every dimension. This implies that eutactic lattices, while always critical points of the packing density function, are not necessarily local maxima or minima even among the well-rounded lattices. We also prove that a (weakly) nearly orthogonal lattice in~$\mathbb R^n$ contains no more than~$4n-2$ minimal vectors (with any smaller even number possible) and establish some bounds on coherence of these lattices., Comment: 21 pages, to appear in Linear Algebra and its Applications

We give a new deterministic construction of integer sensing matrices that can be used for the recovery of integer-valued signals in compressed sensing. This is a family of $n \times d$ integer matrices, $d \geq n$, with bounded sup-norm and the property that no $\ell$ column vectors are linearly dependent, $\ell \leq n$. Further, if $\ell \leq o(\log n)$ then $d/n \to \infty$ as $n \to \infty$. Our construction comes from particular sets of difference vectors of point-sets in $\mathbb R^n$ that cannot be covered by few parallel hyperplanes. We construct examples of such sets on the $0, \pm 1$ grid and use them for the matrix construction. We also show a connection of our constructions to a simple version of the Tarski plank problem., Comment: 11 pages, 1 figure; to appear in Discrete and Computational Geometry

Frobenius problem and its many generalizations have been extensively studied in several areas of mathematics. We study semigroups of totally positive algebraic integers in totally real number fields, defining analogues of the Frobenius numbers in this context. We use a geometric framework recently introduced by Aliev, De Loera and Louveaux to produce upper bounds on these Frobenius numbers in terms of a certain height function. We discuss some properties of this function, relating it to absolute Weil height and obtaining a lower bound in the spirit of Lehmer's conjecture for algebraic vectors satisfying some special conditions. We also use a result of Borosh and Treybig to obtain bounds on the size of representations and number of elements of bounded height in such positive semigroups of totally real algebraic integers., Comment: 13 pages, to appear in Moscow Journal of Combinatorics and Number Theory

Given an isotropic quadratic form over a number field which assumes a value $t$, we investigate the distribution of points at which this value is assumed. Building on the previous work about the distribution of small-height zeros of quadratic forms, we produce bounds on height of points outside of some algebraic sets in a quadratic space at which the form assumes the value $t$. Our bounds on height are explicit in terms of the heights of the form, the space, the algebraic set and the value $t$., Comment: 11 pages; to appear in Journal of Number Theory

We show that real tight frames that generate lattices must be rational, and use this observation to describe a construction of lattices from vertex transitive graphs. In the case of irreducible group frames, we show that the corresponding lattice is always strongly eutactic. This is the case for the more restrictive class of distance transitive graphs. We show that such lattices exist in arbitrarily large dimensions and demonstrate examples arising from some notable families of graphs. In particular, some well-known root lattices and those related to them can be recovered this way. We discuss various properties of this construction and also mention some potential applications of lattices generated by incoherent systems of vectors., Comment: some corrections of typos made; updated bibliography

Ideal lattices in the plane coming from real quadratic number fields have been investigated by several authors in the recent years. In particular, it has been proved that every such ideal has a basis that can be twisted by the action of the diagonal group into a Minkowski reduced basis for a well-rounded lattice. We explicitly study such twists on the canonical bases of ideals, which are especially important in arithmetic theory of quadratic number fields and binary quadratic forms. Specifically, we prove that every fixed real quadratic field has only finitely many ideals whose canonical basis can be twisted into a well-rounded or a stable lattice in the plane. We demonstrate some explicit examples of such twists. We also briefly discuss the relation between stable and well-rounded twists of arbitrary ideal bases., Comment: 16 pages, 3 figures; to appear in the Houston Journal of Mathematics

The Bateman-Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green-Tao theorem, along with many famous conjectures, such the twin prime conjecture and Landau's conjecture. We discuss the Bateman-Horn conjecture, its applications, and its origins., Comment: 47 pages, formerly titled "One conjecture to rule them all: Bateman-Horn," to appear in Expositiones Mathematicae

Let $\Lambda \subset \mathbb R^n$ be an algebraic lattice, coming from a projective module over the ring of integers of a number field $K$. Let $\mathcal Z \subset \mathbb R^n$ be the zero locus of a finite collection of polynomials such that $\Lambda \nsubseteq \mathcal Z$ or a finite union of proper full-rank sublattices of $\Lambda$. Let $K_1$ be the number field generated over $K$ by coordinates of vectors in $\Lambda$, and let $L_1,\dots,L_t$ be linear forms in $n$ variables with algebraic coefficients satisfying an appropriate linear independence condition over $K_1$. For each $\varepsilon > 0$ and $\boldsymbol a \in \mathbb R^n$, we prove the existence of a vector $\boldsymbol x \in \Lambda \setminus \mathcal Z$ of explicitly bounded sup-norm such that $$\| L_i(\boldsymbol x) - a_i \| < \varepsilon$$ for each $1 \leq i \leq t$, where $\|\ \|$ stands for the distance to the nearest integer. The bound on sup-norm of $\boldsymbol x$ depends on $\varepsilon$, as well as on $\Lambda$, $K$, $\mathcal Z$ and heights of linear forms. This presents a generalization of Kronecker's approximation theorem, establishing an effective result on density of the image of $\Lambda \setminus \mathcal Z$ under the linear forms $L_1,\dots,L_t$ in the $t$-torus~$\mathbb R^t/\mathbb Z^t$., Comment: 13 pages, to appear in the Proceedings of AMS

Compressed sensing is a relatively new mathematical paradigm that shows a small number of linear measurements are enough to efficiently reconstruct a large dimensional signal under the assumption the signal is sparse. Applications for this technology are ubiquitous, ranging from wireless communications to medical imaging, and there is now a solid foundation of mathematical theory and algorithms to robustly and efficiently reconstruct such signals. However, in many of these applications, the signals of interest do not only have a sparse representation, but have other structure such as lattice-valued coefficients. While there has been a small amount of work in this setting, it is still not very well understood how such extra information can be utilized during sampling and reconstruction. Here, we explore the problem of integer sparse reconstruction, lending insight into when this knowledge can be useful, and what types of sampling designs lead to robust reconstruction guarantees. We use a combination of combinatorial, probabilistic and number-theoretic methods to discuss existence and some constructions of such sensing matrices with concrete examples. We also prove sparse versions of Minkowski's Convex Body and Linear Forms theorems that exhibit some limitations of this framework.

This paper supplies additions to our paper in Linear Algebra Appl. 510 (2016) 395--420 on integral spans of tight frames in Euclidean spaces. In that previous paper, we considered the case of an equiangular tight frame (ETF), proving that if its integral span is a lattice then the frame must be rational, but overlooking a simple argument in the reverse direction. Thus our first result here is that the integral span of an ETF is a lattice if and only if the frame is rational. Further, we discuss conditions under which such lattices are eutactic and perfect and, consequently, are local maxima of the packing density function in the dimension of their span. In particular, the unit (276, 23) equiangular tight frame is shown to be eutactic and perfect. More general tight frames and their norm-forms are considered as well, and definitive results are obtained in dimensions two and three., Comment: 9 pages; In this version some mistakes occurring in version 1 are corrected

We present a new family of one-coincidence sequence sets suitable for frequency hopping code division multiple access (FH-CDMA) systems with dispersed (low density) sequence elements. These sets are derived from one-coincidence prime sequence sets, such that for each one-coincidence prime sequence set there is a new one-coincidence set comprised of sequences with dispersed sequence elements, required in some circumstances, for FH-CDMA systems. Getting rid of crowdedness of sequence elements is achieved by doubling the size of the sequence element alphabet. In addition, this doubling process eases control over the distance between adjacent sequence elements. Properties of the new sets are discussed., Comment: 8 pages, 5 tables; to appear in Advances in Mathematics of Communication

We investigate properties of zeta functions of polynomial rings and their quotients, generalizing and extending some classical results about Dedekind zeta functions of number fields. By an application of Delange's version of the Ikehara Tauberian Theorem, we are then able to determine the asymptotic order of the ideal counting function in such rings. As a result, we produce counting estimates on ideal lattices of bounded determinant coming from fixed number fields, as well as density estimates for any ideal lattices among all sublattices of $\mathbb Z^d$. We conclude with some more general speculations and open questions., Comment: 11 pages

We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular $(k,n)$ frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the $k$-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for $n = k+1$ and that there are infinitely many $k$ such that a lattice emerges for $n = 2k$. We dispose of all cases in dimensions $k$ at most $9$. In particular, we show that a $(7,28)$ frame generates a strongly eutactic lattice and give an alternative proof of Roland Bacher's recent observation that this lattice is perfect., Comment: 25 pages

We investigate similarity classes of arithmetic lattices in the plane. We introduce a natural height function on the set of such similarity classes, and give asymptotic estimates on the number of all arithmetic similarity classes, semi-stable arithmetic similarity classes, and well-rounded arithmetic similarity classes of bounded height as the bound tends to infinity. We also briefly discuss some properties of the $j$-invariant corresponding to similarity classes of planar lattices., Comment: 13 pages; to appear in Proc. Amer. Math. Soc

We consider lattices generated by finite Abelian groups. We prove that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a uniform normalized tight frame. Further, our result combined with a recent theorem of R. Bacher produces (via the classical Voronoi criterion) a new infinite family of extreme lattices. Additionally, we investigate the structure of the automorphism groups of these lattices, strengthening our previous results in this direction., Comment: 12 pages

In this paper we establish three results on small-height zeros of quadratic polynomials over $\overline{\mathbb Q}$. For a single quadratic form in $N \geq 2$ variables on a subspace of $\overline{\mathbb Q}^N$, we prove an upper bound on the height of a smallest nontrivial zero outside of an algebraic set under the assumption that such a zero exists. For a system of $k$ quadratic forms on an $L$-dimensional subspace of $\overline{\mathbb Q}^N$, $N \geq L \geq \frac{k(k+1)}{2}+1$, we prove existence of a nontrivial simultaneous small-height zero. For a system of one or two inhomogeneous quadratic and $m$ linear polynomials in $N \geq m+4$ variables, we obtain upper bounds on the height of a smallest simultaneous zero, if such a zero exists. Our investigation extends previous results on small zeros of quadratic forms, including Cassels' theorem and its various generalizations and contributes to the literature of so-called "absolute" Diophantine results with respect to height. All bounds on height are explicit., Comment: 23 pages; to appear in Research in Mathematical Sciences. arXiv admin note: substantial text overlap with arXiv:1307.0564

We consider the well-known Rosenbloom-Tsfasman function field lattices in the special case of Hermitian function fields. We show that in this case the resulting lattices are generated by their minimal vectors, provide an estimate on the total number of minimal vectors, and derive properties of the automorphism groups of these lattices. Our study continues previous investigations of lattices coming from elliptic curves and finite Abelian groups. The lattices we are faced with here are more subtle than those considered previously, and the proofs of the main results require the replacement of the existing linear algebra approaches by deep results of Gerhard Hiss on the factorization of functions with particular divisor support into lines and their inverses., Comment: 15 pages

Let $K$ be a global field or $\overline{\mathbb Q}$, $F$ a nonzero quadratic form on $K^N$, $N \geq 2$, and $V$ a subspace of $K^N$. We prove the existence of an infinite collection of finite families of small-height maximal totally isotropic subspaces of $(V,F)$ such that each such family spans $V$ as a $K$-vector space. This result generalizes and extends a well known theorem of J. Vaaler and further contributes to the effective study of quadratic forms via height in the general spirit of Cassels' theorem on small zeros of quadratic forms. All bounds on height are explicit., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1307.0564

We say that a Euclidean lattice in $\mathbb R^n$ is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group $S_n$, i.e., if the lattice is closed under the action of some non-identity elements of $S_n$. Given a fixed element $\tau \in S_n$, we study properties of the set of all lattices closed under the action of $\tau$: we call such lattices $\tau$-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio, which we studied in a recent paper. Continuing our investigation, we discuss some basic properties of permutation invariant lattices, in particular proving that the subset of well-rounded lattices in the set of all $\tau$-invariant lattices in $\mathbb R^n$ has positive co-dimension (and hence comprises zero proportion) for all $\tau$ different from an $n$-cycle., Comment: corrected Lemma 2.1

This paper is devoted to the study of lattices generated by finite Abelian groups. Special species of such lattices arise in the exploration of elliptic curves over finite fields. In case the generating group is cyclic, they are also known as the Barnes lattices. It is shown that for every finite Abelian group with the exception of the cyclic group of order four these lattices have a basis of minimal vectors. Another result provides an improvement of a recent upper bound by Min Sha for the covering radius in the case of the Barnes lattices. Also discussed are properties of the automorphism groups of these lattices., Comment: 26 pages

The paper is devoted to exact and asymptotic formulas for the determinants of Toeplitz matrices with perturbations by blocks of fixed size in the four corners. If the norms of the inverses of the unperturbed matrices remain bounded as the matrix dimension goes to infinity, then standard perturbation theory yields asymptotic expressions for the perturbed determinants. This premise is not satisfied for matrices generated by so-called Fisher-Hartwig symbols. In that case we establish formulas for pure single Fisher-Hartwig singularities and for Hermitian matrices induced by general Fisher-Hartwig symbols., Comment: 20 pages

Cyclic lattices are sublattices of $\mathbb Z^N$ that are preserved under the rotational shift operator. Cyclic lattices were introduced by D.~Micciancio and their properties were studied in the recent years by several authors due to their importance in cryptography. In particular, Peikert and Rosen showed that on cyclic lattices in prime dimensions, the shortest independent vectors problem SIVP reduces to the shortest vector problem SVP with a particularly small loss in approximation factor, as compared to general lattices. In this paper, we further investigate geometric properties of cyclic lattices. Our main result is a counting estimate for the number of well-rounded cyclic lattices, indicating that well-rounded lattices are more common among cyclic lattices than generically. We also show that SVP is equivalent to SIVP on a positive proportion of Minkowskian well-rounded cyclic lattices in every dimension. As an example, we demonstrate an explicit construction of a family of such lattices on which this equivalence holds. To conclude, we introduce a class of sublattices of $\mathbb Z^N$ closed under the action of subgroups of the permutation group $S_N$, which are a natural generalization of cyclic lattices, and show that our results extend to all such lattices closed under the action of any $N$-cycle., Comment: A mistake in Theorems 1.1 and 1.2 is corrected

We study semi-stable ideal lattices coming from real quadratic number fields. Specifically, we demonstrate infinite families of semi-stable and unstable ideal lattices of trace type, establishing explicit conditions on the canonical basis of an ideal that ensure stability; in particular, our result implies that an ideal lattice of trace type coming from a real quadratic field is semi-stable with positive probability. We also briefly discuss the connection between stability and well-roundedness of Euclidean lattices., Comment: Theorem 1.2 of the previous version was incorrect as stated, it is now removed

In their well known book Tsfasman and Vladut introduced a construction of a family of function field lattices from algebraic curves over finite fields, which have asymptotically good packing density in high dimensions. In this paper we study geometric properties of lattices from this construction applied to elliptic curves. In particular, we determine the generating sets, conditions for well-roundedness and a formula for the number of minimal vectors. We also prove a bound on the covering radii of these lattices, which improves on the standard inequalities., Comment: 10 pages, to appear in Finite Fields and Their Applications; some minor revisions and improvements in this version

An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit applications of a particular estimate of this sort to several counting problems in number theory: counting integral points and units of bounded height over number fields, counting points of bounded height over positive definite quaternion algebras, and counting points of bounded height with a fixed support over global function fields. Our arguments use a collection of height comparison inequalities for heights over a number field and over a quaternion algebra. We also show how these inequalities can be used to obtain existence results for points of bounded height over a quaternion algebra, which constitute non-commutative analogues of variations of the classical Siegel's lemma and Cassels' theorem on small zeros of quadratic forms., Comment: 22 pages; to appear in the Online Journal of Analytic Combinatorics

Let $F$ be a quadratic form in $N \geq 2$ variables defined on a vector space $V \subseteq K^N$ over a global field $K$, and $\Z \subseteq K^N$ be a finite union of varieties defined by families of homogeneous polynomials over $K$. We show that if $V \setminus \Z$ contains a nontrivial zero of $F$, then there exists a linearly independent collection of small-height zeros of $F$ in $V\setminus \Z$, where the height bound does not depend on the height of $\Z$, only on the degrees of its defining polynomials. As a corollary of this result, we show that there exists a small-height maximal totally isotropic subspace $W$ of the quadratic space $(V,F)$ such that $W$ is not contained in $\Z$. Our investigation extends previous results on small zeros of quadratic forms, including Cassels' theorem and its various generalizations. The paper also contains an appendix with two variations of Siegel's lemma. All bounds on height are explicit., Comment: 25 pages; to appear in Transactions of AMS

We study well-rounded lattices which come from ideals in quadratic number fields, generalizing some recent results of the first author with K. Petersen. In particular, we give a characterization of ideal well-rounded lattices in the plane and show that a positive proportion of real and imaginary quadratic number fields contains ideals giving rise to well-rounded lattices., Comment: 13 pages; to appear in the International Journal of Number Theory

We investigate distribution of integral well-rounded lattices in the plane, parameterizing the set of their similarity classes by solutions of the family of Pell-type Diophantine equations of the form $x^2+Dy^2=z^2$ where $D>0$ is squarefree. We apply this parameterization to the study of the greatest minimal norm and the highest signal-to-noise ratio on the set of such lattices with fixed determinant, also estimating cardinality of these sets (up to rotation and reflection) for each determinant value. This investigation extends previous work of the first author in the specific cases of integer and hexagonal lattices and is motivated by the importance of integral well-rounded lattices for discrete optimization problems. We briefly discuss an application of our results to planar lattice transmitter networks., Comment: 12 pages, to appear in Discrete and Computational Geometry

We investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at $s=1$ with a real pole of order 2, improving upon a recent result of S. Kuehnlein. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less or equal $N$ is $O(N \log N)$ as $N \to \infty$. To obtain these results, we produce a description of integral well-rounded sublattices of a fixed planar integral well-rounded lattice and investigate convergence properties of a zeta-function of similarity classes of such lattices, building on some previous results of the author., Comment: 12 pages; to appear in PAMS