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Colloquium of the Steklov Mathematical Institute of Russian Academy of Sciences
December 1, 2016 16:00, Moscow, Steklov Mathematical Institute of RAS, Conference Hall (8 Gubkina)

Simple questions about arrangement of points on hiher-dimensional sphere and cube, answers for which are not known

G. A. Kabatiansky
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G. A. Kabatiansky
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Abstract: Let $M$ be any metric space with distance $d(x,y)$. Define the function $A(M,s)$ to be the maximal number of points in $M$ such that all pairwise distances between points are not less than $s$, where $s>0$. We consider as $M$ the unit sphere in the $n$-dimensional Euclidean space and the cube with vertices whose coordinates are $+1$ or $-1$, in the same space. In the case of the unit sphere it is convenient to replace Euclidean distance by the angular distance, and in the case of the cube by the number of coordinates that are different for two given vectors, which is called a Hamming distance. The first problem is studied by discrete geometry, and the second one by the coding theory. We will show some similarity between these two problems. In particular, in the both cases the function $A(M,s)$ behaves “in a threshold manner,” i.e., if s greater than the threshold, then $A(M,s)$ does not increase more than linearly with $n$, and if $s$ is smaller then $A(M,s)$ grows exponentially. The threshold is 90 degrees for the sphere and is $n/2$ for the cube. However, the order of the exponent is not known, only the upper and lower bounds are known. Nevertheless these bounds give, in particular, the best upper and lower asymptotic bounds for the kissing number, as well as the fact that the sphere packing density of the Euclidean $n$-dimensional space lies (asymptotically) between $2^{-n}$ and $2^{-0.599n}$.

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