The smallest sets of points not determined by their X-rays

TL;DR

This paper investigates the minimal size of point sets in Euclidean and integer spaces that are indistinguishable by their X-ray projections in multiple directions, providing new polynomial bounds and implications for classical number theory problems.

Contribution

The authors establish the first polynomial upper bounds on the size of point sets with identical X-rays in multiple directions in higher dimensions, extending previous exponential bounds.

Findings

Proved $oxed{ ext{ $ ext{O}(m^{d+1+ ext{ε}})$}}$ bounds for $oxed{ ext{ $ ext{ψ}_{ extbf{K}^d}(m)$}}$ in higher dimensions.

Derived bounds on solutions to the Prouhet-Tarry-Escott problem.

Established lower bounds leading to a strengthened Rényi's theorem for integer lattice points.

Abstract

Let $F$ be an $n$-point set in $\mathbb{K}^d$ with $\mathbb{K}\in\{\mathbb{R},\mathbb{Z}\}$ and $d\geq 2$. A (discrete) X-ray of $F$ in direction $s$ gives the number of points of $F$ on each line parallel to $s$. We define $\psi_{\mathbb{K}^d}(m)$ as the minimum number $n$ for which there exist $m$ directions $s_1,...,s_m$ (pairwise linearly independent and spanning $\mathbb{R}^d$) such that two $n$-point sets in $\mathbb{K}^d$ exist that have the same X-rays in these directions. The bound $\psi_{\mathbb{Z}^d}(m)\leq 2^{m-1}$ has been observed many times in the literature. In this note we show $\psi_{\mathbb{K}^d}(m)=O(m^{d+1+\varepsilon})$ for $\varepsilon>0$. For the cases $\mathbb{K}^d=\mathbb{Z}^d$ and $\mathbb{K}^d=\mathbb{R}^d$, $d>2$, this represents the first upper bound on $\psi_{\mathbb{K}^d}(m)$ that is polynomial in $m$. As a corollary we derive bounds on the sizes of solutions to both the classical and two-dimensional Prouhet-Tarry-Escott problem. Additionally, we establish lower bounds on $\psi_{\mathbb{K}^d}$ that enable us to prove a strengthened version of R\'enyi's theorem for points in $\mathbb{Z}^2$.