How many sorted projections determine n points in k dimensions?

Abstract

Let $d_1,\dots ,d_m$ be directions in ${\mathbb{R}}^k$, where $k\ge 2$. A multiset $X$ of $n$ points of ${\mathbb{R}}^k$ projects to the $m$ multisets of numbers $\{\{x\cdot d_j:x\in X\}\}$, or equivalently to the $m$ sorted lists of these numbers. How many directions are needed for these projections to determine every multiset of $n$ points? Two simple bounds are known in every dimension: more than $(k-1){\log }_{2}n$ directions are always needed, and $(k-1)n+1$ directions in general position always suffice. In the plane much more is known: Matousek, Privetivy and Skovron showed that almost every choice of $O(\log n\log \log n)$ directions suffices. Whether anything similar holds in higher dimensions was left open by Dym, Wellershoff, Tsoukanis, Levy and Balan. We observe that two lemmas of Hair and Sahai, proved for list recovery of random linear codes, answer that question, and by tightening their argument we show that in every dimension almost every choice of $(4k-2)\ln (2n)\ln (k{\log }_{2}2n)+k+1$ directions suffices. For $k\le 10$ this beats the linear bound once $n$ is larger than about 100: for example, 144 directions suffice for multisets of 1000 points in the plane, and an exact computation lowers this to 133. Letting some of the random blocks in their argument fail removes a factor $\log k$: almost every choice of $O(k\log n\log \log n)$ directions suffices, uniformly in $k$ and $n$. We also show that $2k-1$ directions never determine three points, so that the logarithmic lower bound is not tight; that every $m$ directions in the plane fail on some pair of multisets of size $O(1.78^m)$, improving the bound $O(1.80^m)$ of Matousek, Privetivy and Skovron; and that for each $k$, $m$ and $n$, either almost every choice of $m$ directions works or almost none does.

arXiv comment28 pages, 5 figures