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1. CMB 2011 (vol 56 pp. 354)

Hare, Kathryn E.; Mendivil, Franklin; Zuberman, Leandro
The Sizes of Rearrangements of Cantor Sets
A linear Cantor set $C$ with zero Lebesgue measure is associated with the countable collection of the bounded complementary open intervals. A rearrangment of $C$ has the same lengths of its complementary intervals, but with different locations. We study the Hausdorff and packing $h$-measures and dimensional properties of the set of all rearrangments of some given $C$ for general dimension functions $h$. For each set of complementary lengths, we construct a Cantor set rearrangement which has the maximal Hausdorff and the minimal packing $h$-premeasure, up to a constant. We also show that if the packing measure of this Cantor set is positive, then there is a rearrangement which has infinite packing measure.

Keywords:Hausdorff dimension, packing dimension, dimension functions, Cantor sets, cut-out set
Categories:28A78, 28A80

2. CMB 2011 (vol 54 pp. 645)

Flores, André Luiz; Interlando, J. Carmelo; Neto, Trajano Pires da Nóbrega
An Extension of Craig's Family of Lattices
Let $p$ be a prime, and let $\zeta_p$ be a primitive $p$-th root of unity. The lattices in Craig's family are $(p-1)$-dimensional and are geometrical representations of the integral $\mathbb Z[\zeta_p]$-ideals $\langle 1-\zeta_p \rangle^i$, where $i$ is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions $p-1$ where $149 \leq p \leq 3001$, Craig's lattices are the densest packings known. Motivated by this, we construct $(p-1)(q-1)$-dimensional lattices from the integral $\mathbb Z[\zeta _{pq}]$-ideals $\langle 1-\zeta_p \rangle^i \langle 1-\zeta_q \rangle^j$, where $p$ and $q$ are distinct primes and $i$ and $j$ are positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties.

Keywords:geometry of numbers, lattice packing, Craig's lattices, quadratic forms, cyclotomic fields
Categories:11H31, 11H55, 11H50, 11R18, 11R04

3. CMB 2009 (vol 52 pp. 407)

Lángi, Zsolt; Naszódi, Márton
On the Bezdek--Pach Conjecture for Centrally Symmetric Convex Bodies
The Bezdek--Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in $\Re^d$ is $2^d$. Nasz\'odi proved that the quantity in question is not larger than $2^{d+1}$. We present an improvement to this result by proving the upper bound $3\cdot2^{d-1}$ for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies.

Keywords:Bezdek--Pach Conjecture, homothets, packing, Hadwiger number, antipodality
Categories:52C17, 51N20, 51K05, 52A21, 52A37

4. CMB 2002 (vol 45 pp. 634)

Lagarias, Jeffrey C.; Pleasants, Peter A. B.
Local Complexity of Delone Sets and Crystallinity
This paper characterizes when a Delone set $X$ in $\mathbb{R}^n$ is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the heterogeneity of their distribution. For a Delone set $X$, let $N_X (T)$ count the number of translation-inequivalent patches of radius $T$ in $X$ and let $M_X(T)$ be the minimum radius such that every closed ball of radius $M_X(T)$ contains the center of a patch of every one of these kinds. We show that for each of these functions there is a ``gap in the spectrum'' of possible growth rates between being bounded and having linear growth, and that having sufficiently slow linear growth is equivalent to $X$ being an ideal crystal. Explicitly, for $N_X(T)$, if $R$ is the covering radius of $X$ then either $N_X(T)$ is bounded or $N_X (T) \ge T/2R$ for all $T>0$. The constant $1/2R$ in this bound is best possible in all dimensions. For $M_X(T)$, either $M_X(T)$ is bounded or $M_X(T)\ge T/3$ for all $T>0$. Examples show that the constant $1/3$ in this bound cannot be replaced by any number exceeding $1/2$. We also show that every aperiodic Delone set $X$ has $M_X(T)\ge c(n) T$ for all $T>0$, for a certain constant $c(n)$ which depends on the dimension $n$ of $X$ and is $>1/3$ when $n>1$.

Keywords:aperiodic set, Delone set, packing-covering constant, sphere packing
Categories:52C23, 52C17

5. CMB 1999 (vol 42 pp. 380)

Nurmela, Kari J.; Östergård, Patric R. J.; aus dem Spring, Rainer
Asymptotic Behavior of Optimal Circle Packings in a Square
A lower bound on the number of points that can be placed in a square of side $\sigma$ such that no two points are within unit distance from each other is proven. The result is constructive, and the series of packings obtained contains many conjecturally optimal packings.

Keywords:asymptotic bound, circle packing

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