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Search: MSC category 52C17 ( Packing and covering in $n$ dimensions [See also 05B40, 11H31] )

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1. CJM 2011 (vol 64 pp. 1036)

Koh, Doowon; Shen, Chun-Yen
Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields
In this paper we study the extension problem, the averaging problem, and the generalized Erdős-Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial.

Keywords:extension problems, averaging operator, finite fields, Erdős-Falconer distance problems, homogeneous polynomial
Categories:42B05, 11T24, 52C17

2. CJM 1998 (vol 50 pp. 16)

Böröczky, Károly; Schnell, Uwe
Asymptotic shape of finite packings
Let $K$ be a convex body in $\ed$ and denote by $\cn$ the set of centroids of $n$ non-overlapping translates of $K$. For $\varrho>0$, assume that the parallel body $\cocn+\varrho K$ of $\cocn$ has minimal volume. The notion of parametric density (see~\cite{Wil93}) provides a bridge between finite and infinite packings (see~\cite{BHW94} or~\cite{Hen}). It is known that there exists a maximal $\varrho_s(K)\geq 1/(32d^2)$ such that $\cocn$ is a segment for $\varrho<\varrho_s$ (see~\cite{BHW95}). We prove the existence of a minimal $\varrho_c(K)\leq d+1$ such that if $\varrho>\varrho_c$ and $n$ is large then the shape of $\cocn$ can not be too far from the shape of $K$. For $d=2$, we verify that $\varrho_s=\varrho_c$. For $d\geq 3$, we present the first example of a convex body with known $\varrho_s$ and $\varrho_c$; namely, we have $\varrho_s=\varrho_c=1$ for the parallelotope.

Categories:52C17, 05B40

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