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Let
$\mathbf{p}$
be a configuration of
$n$
points in
$\mathbb{R}^{d}$
for some
$n$
and some
$d\geqslant 2$
. Each pair of points has a Euclidean distance in the configuration. Given some graph
$G$
on
$n$
vertices, we measure the point-pair distances corresponding to the edges of
$G$
. In this paper, we study the question of when a generic
$\mathbf{p}$
in
$d$
dimensions will be uniquely determined (up to an unknowable Euclidean transformation) from a given set of point-pair distances together with knowledge of
$d$
and
$n$
. In this setting the distances are given simply as a set of real numbers; they are not labeled with the combinatorial data that describes which point pair gave rise to which distance, nor is data about
$G$
given. We show, perhaps surprisingly, that in terms of generic uniqueness, labels have no effect. A generic configuration is determined by an unlabeled set of point-pair distances (together with
$d$
and
$n$
) if and only if it is determined by the labeled distances.
We generalize work by Bourgain and Kontorovich [On the local-global conjecture for integral Apollonian gaskets, Invent. Math. 196 (2014), 589–650] and Zhang [On the local-global principle for integral Apollonian 3-circle packings, J. Reine Angew. Math. 737, (2018), 71–110], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group
${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$
satisfying certain conditions, where
$K$
is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof is that
${\mathcal{A}}$
possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in
$\operatorname{PSL}_{2}({\mathcal{O}}_{K})$
containing a Zariski dense subgroup of
$\operatorname{PSL}_{2}(\mathbb{Z})$
.
For each
$n\geqslant 2$
we construct a measurable subset of the unit ball in
$\mathbb{R}^{n}$
that does not contain pairs of points at distance 1 and whose volume is greater than
$(1/2)^{n}$
times the volume of the unit ball. This disproves a conjecture of Larman and Rogers from 1972.
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.
We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known: they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. For all such spaces the best possible bounds for the quadratic discrepancies and sums of pairwise distances are obtained in the paper (Theorems 2.1 and 2.2). Distributions of points of
$t$
-designs on such spaces are also considered (Theorem 2.3). In particular, it is shown that the optimal
$t$
-designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances (Corollary 2.1). Our approach is based on the Fourier analysis on two-point homogeneous spaces and explicit spherical function expansions for discrepancies and sums of distances (Theorems 4.1 and 4.2).
the pioneer of interchange laws in universal algebra
We establish a combinatorial model for the Boardman–Vogt tensor product of several absolutely free operads, that is, free symmetric operads that are also free as 𝕊-modules. Our results imply that such a tensor product is always a free 𝕊-module, in contrast with the results of Kock and Bremner–Madariaga on hidden commutativity for the Boardman–Vogt tensor square of the operad of non-unital associative algebras.
We prove a lower bound on the entropy of sphere packings of
$\mathbb{R}^{d}$
of density
$\unicode[STIX]{x1D6E9}(d\cdot 2^{-d})$
. The entropy measures how plentiful such packings are, and our result is significantly stronger than the trivial lower bound that can be obtained from the mere existence of a dense packing. Our method also provides a new, statistical-physics-based proof of the
$\unicode[STIX]{x1D6FA}(d\cdot 2^{-d})$
lower bound on the maximum sphere packing density by showing that the expected packing density of a random configuration from the hard sphere model is at least
$(1+o_{d}(1))\log (2/\sqrt{3})d\cdot 2^{-d}$
when the ratio of the fugacity parameter to the volume covered by a single sphere is at least
$3^{-d/2}$
. Such a bound on the sphere packing density was first achieved by Rogers, with subsequent improvements to the leading constant by Davenport and Rogers, Ball, Vance, and Venkatesh.
We study local properties of the Bakry–Émery curvature function
${\mathcal{K}}_{G,x}:(0,\infty ]\rightarrow \mathbb{R}$
at a vertex
$x$
of a graph
$G$
systematically. Here
${\mathcal{K}}_{G,x}({\mathcal{N}})$
is defined as the optimal curvature lower bound
${\mathcal{K}}$
in the Bakry–Émery curvature-dimension inequality
$CD({\mathcal{K}},{\mathcal{N}})$
that
$x$
satisfies. We provide upper and lower bounds for the curvature functions, introduce fundamental concepts like curvature sharpness and
$S^{1}$
-out regularity, and relate the curvature functions of
$G$
with various spectral properties of (weighted) graphs constructed from local structures of
$G$
. We prove that the curvature functions of the Cartesian product of two graphs
$G_{1},G_{2}$
are equal to an abstract product of curvature functions of
$G_{1},G_{2}$
. We explore the curvature functions of Cayley graphs and many particular (families of) examples. We present various conjectures and construct an infinite increasing family of 6-regular graphs which satisfy
$CD(0,\infty )$
but are not Cayley graphs.
In this paper we discuss some dimension results for triangle sets of compact sets in
$\mathbb{R}^{2}$
. In particular we prove that for any compact set
$F$
in
$\mathbb{R}^{2}$
, the triangle set
$\unicode[STIX]{x1D6E5}(F)$
satisfies
In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results. Except for parallelograms and centrally symmetric hexagons, there are no other convex domains that can form two-, three- or four-fold lattice tilings in the Euclidean plane. However, there are both octagons and decagons that can form five-fold lattice tilings. Whenever
$n\geqslant 3$
, there are non-parallelohedral polytopes that can form five-fold lattice tilings in the
$n$
-dimensional Euclidean space.
We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.
It was shown by Gruslys, Leader and Tan that any finite subset of
$\mathbb{Z}^{n}$
tiles
$\mathbb{Z}^{d}$
for some
$d$
. The first non-trivial case is the punctured interval, which consists of the interval
$\{-k,\ldots ,k\}\subset \mathbb{Z}$
with its middle point removed: they showed that this tiles
$\mathbb{Z}^{d}$
for
$d=2k^{2}$
, and they asked if the dimension needed tends to infinity with
$k$
. In this note we answer this question: we show that, perhaps surprisingly, every punctured interval tiles
$\mathbb{Z}^{4}$
.
In the realm of Delone sets in locally compact, second countable Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty–Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topological limits: we show that the measured quantities under consideration can be approximated via periodic analogs.
Let C be a bounded convex object in ℝd, and let P be a set of n points lying outside C. Further, let cp, cq be two integers with 1 ⩽ cq ⩽ cp ⩽ n - ⌊d/2⌋, such that every cp + ⌊d/2⌋ points of P contain a subset of size cq + ⌊d/2⌋ whose convex hull is disjoint from C. Then our main theorem states the existence of a partition of P into a small number of subsets, each of whose convex hulls are disjoint from C. Our proof is constructive and implies that such a partition can be computed in polynomial time.
In particular, our general theorem implies polynomial bounds for Hadwiger--Debrunner (p, q) numbers for balls in ℝd. For example, it follows from our theorem that when p > q = (1+β)⋅d/2 for β > 0, then any set of balls satisfying the (p, q)-property can be hit by O((1+β)2d2p1+1/β logp) points. This is the first improvement over a nearly 60 year-old exponential bound of roughly O(2d).
Our results also complement the results obtained in a recent work of Keller, Smorodinsky and Tardos where, apart from improvements to the bound on HD(p, q) for convex sets in ℝd for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane.
Given any positive integers
$m$
and
$d$
, we say a sequence of points
$(x_{i})_{i\in I}$
in
$\mathbb{R}^{m}$
is Lipschitz-
$d$
-controlling if one can select suitable values
$y_{i}\;(i\in I)$
such that for every Lipschitz function
$f\,:\,\mathbb{R}^{m}\,\rightarrow \,\mathbb{R}^{d}$
there exists
$i$
with
$|f(x_{i})\,-\,y_{i}|\,<\,1$
. We conjecture that for every
$m\leqslant d$
, a sequence
$(x_{i})_{i\in I}\subset \mathbb{R}^{m}$
is
$d$
-controlling if and only if
We prove that this condition is necessary and a slightly stronger one is already sufficient for the sequence to be
$d$
-controlling. We also prove the conjecture for
$m=1$
.
A set of points in d-dimensional Euclidean space is almost equidistant if, among any three points of the set, some two are at distance 1. We show that an almost-equidistant set in ℝd has cardinality O(d4/3).
We study a combinatorial problem that recently arose in the context of shape optimization: among all triangles with vertices
$(0,0)$
,
$(x,0)$
, and
$(0,y)$
and fixed area, which one encloses the most lattice points from
$\mathbb{Z}_{{>}0}^{2}$
? Moreover, does its shape necessarily converge to the isosceles triangle
$(x=y)$
as the area becomes large? Laugesen and Liu suggested that, in contrast to similar problems, there might not be a limiting shape. We prove that the limiting set is indeed non-trivial and contains infinitely many elements. We also show that there exist “bad” areas where no triangle is particularly good at capturing lattice points and show that there exists an infinite set of slopes
$y/x$
such that any associated triangle captures more lattice points than any other fixed triangle for infinitely many (and arbitrarily large) areas; this set of slopes is a fractal subset of
$[1/3,3]$
and has Minkowski dimension of at most
$3/4$
.
We consider finite point subsets (distributions) in compact metric spaces. In the case of general rectifiable metric spaces, non-trivial bounds for sums of distances between points of distributions and for discrepancies of distributions in metric balls are given (Theorem 1.1). We generalize Stolarsky’s invariance principle to distance-invariant spaces (Theorem 2.1). For arbitrary metric spaces, we prove a probabilistic invariance principle (Theorem 3.1). Furthermore, we construct equal-measure partitions of general rectifiable compact metric spaces into parts of small average diameter (Theorem 4.1).
We prove that there is a set F in the plane so that the distance between any two points of F is at least 1, and for any positive ϵ < 1, and every line segment in the plane of length at least ϵ−1−o(1), there is a point of F within distance ϵ of the segment. This is tight up to the o(1)-term in the exponent, improving earlier estimates of Peres, of Solomon and Weiss, and of Adiceam.
Answering a question of Füredi and Loeb [On the best constant for the Besicovitch covering theorem. Proc. Amer. Math. Soc.121(4) (1994), 1063–1073], we show that the maximum number of pairwise intersecting homothets of a
$d$
-dimensional centrally symmetric convex body
$K$
, none of which contains the center of another in its interior, is at most
$O(3^{d}d\log d)$
. If
$K$
is not necessarily centrally symmetric and the role of its center is played by its centroid, then the above bound can be replaced by
$O(3^{d}\binom{2d}{d}d\log d)$
. We establish analogous results for the case where the center is defined as an arbitrary point in the interior of
$K$
. We also show that, in the latter case, one can always find families of at least
$\unicode[STIX]{x1D6FA}((2/\sqrt{3})^{d})$
translates of
$K$
with the above property.