Few distance sets in \(\ell_p\) spaces and \(\ell_p\) product spaces (Q2122671)

Given a metric space \(X\), \(e_{s}(X)\) denotes the maximum number of points such that the pairwise distances take only \(s\) values. The notation \(e(X)\) denotes \(e_{1}(X)\). A set of points \(S\subseteq X\) such that the cardinality of the set \[ \left\{ d(x,y):x,y\in S,x\neq y\right\} \] is \(s\) is called an \(s\)-\textit{distance set}. A \(1\)-distance set is known as an \textit{equilateral set}. It was observed independently by \textit{E. Bannai} et al. [Combinatorica 3, 147--152 (1983; Zbl 0522.05022)] and \textit{A. Blokhuis} [Few-distance sets. CWI Tracts, 7. Centrum voor Wiskunde en Informatica. Amsterdam: Mathematisch Centrum. (1984; Zbl 0548.51014)] that the upper bound on an \(s\)-distance set in \(n\)-dimensional Euclidean space is \(\binom{n+s}{s}\). Similar results have been obtained in the hyperbolic space [\textit{A. Blokhuis}, Few-distance sets. CWI Tracts, 7. Centrum voor Wiskunde en Informatica. Amsterdam: Mathematisch Centrum. (1984; Zbl 0548.51014)], the Hamming space [\textit{O. R. Musin} and \textit{H. Nozaki}, Eur. J. Comb. 32, No. 8, 1182--1190 (2011; Zbl 1235.51020)], and the Johnson space [\textit{O. R. Musin} and \textit{H. Nozaki}, Eur. J. Comb. 32, No. 8, 1182--1190 (2011; Zbl 1235.51020)]. This paper first studies \(s\)-distance sets on the normed space \[ l_{p}^{n}=(\mathbb{R}^{n},\left\Vert \cdot\right\Vert _{p}). \] \textit{N. Alon} and \textit{P. Pudlák} [Geom. Funct. Anal. 13, No. 3, 467--482 (2003; Zbl 1034.46015)] have established Theorem. For every \(p\geq1\), we have \[ e(l_{p}^{n})\leq c_{p}n^{(2p+2/(2p-1))} \] where one may take \(c_{p}=cp\) for an absolute \(c>0\). The first result in this paper is an improvement of the above theorem when \(p\) is large in terms of \(n\). Theorem. Let \(c>0\) be the constant from the above theorem. If \(n>1\) and \(p\geq c(n\log\,n)^{2}\), then we have \[ e(l_{p}^{n})\leq2(p+1)n \] If \(p\) is an even integer, \textit{K. J. Swanepoel} [Arch. Math. 83, No. 2, 164--170 (2004; Zbl 1062.52017)] used a linear independence argument to show that \[ e(l_{p}^{n})\leq \begin{cases}(\frac{p}{2}-1)n+1&\text{ if }p\equiv0\pmod{4} \\ \frac{p}{2}n+1&\text{ if }p\equiv2\pmod{4}. \end{cases} \] The next result in this paper is a generalization of Alon and Pudlák's [loc. cit.] theorem to s-distance sets, which is strictly stronger than Conjecture 5 in [\textit{C. Smyth}, in: Thirty essays on geometric graph theory. Berlin: Springer. 483--487 (2013; Zbl 1276.52020)]. Theorem. If \(s\) is a positive integer and \(p\) is a real number abiding by \(2p>s\), then we have \[ e(l_{p}^{n})\leq c_{p,s}n^{(2ps+2s)/(2p-s)} \] for a constant \(c_{p,s}\), depending on \(p\) and \(s\). The remaining three results in this paper are on equilateral sets in the \(l_{p}\) sum of Euclidean spaces. \textit{K. J. Swanepoel} [Bolyai Soc. Math. Stud. 27, 407--458 (2018; Zbl 1419.52018)] showed that \[ e(X\oplus_{\infty}Y)\leq e(X)b_{f}(Y) \] where \(b_{f}\) is the finite Borsuk number. Theorem. Let \(\mathbb{E}^{a}\) and \(\mathbb{E}^{b}\) be Euclidean spaces. Then we have \[ e(\mathbb{E}^{a}\oplus_{\infty}\mathbb{E}^{b})\leq(a+1)(b+1)+1 \] The next result in this paper provides an upper bound when \(p\) is even. Theorem. Let \(\mathbb{E}^{a}\) and \(\mathbb{E}^{b}\) be Euclidean spaces, and let \(p\) be an even integer. Then we have \[ e(\mathbb{E}^{a}\oplus_{p}\mathbb{E}^{b})\leq\binom{a+p/2}{a}+\binom{b+p/2}{b} \] The final result in this paper is to extend Alon and Pudlák's [loc. cit.] theorem on equilateral sets in \(l_{p}^{n}\) to certain \(l_{p}\) product spaces, presenting an upper bound on the \(l_{p}\) sum of \(n\) Euclidean spaces for any \(1\leq p<\infty\). Theorem. Let \(\mathbb{E}^{a_{1}},\dots,\mathbb{E}^{a_{n}}\) be Euclidean spaces and set \(a=\max_{1\leq i\leq n}a_{i}\). If \(2p>a\), then we have \[ e(\mathbb{E}^{a_{1}}\oplus_{p}\cdots\oplus_{p}\mathbb{E}^{a_{n} })\leq c_{p,a}n^{\frac{2p+2a}{2p-a}} \] A synopsis of the paper goes as follows. \S 2 presents two famous results to be used in this paper, the first being the rank lemma which allows of estimating the rank of a symmetric matrix, while the second being Jackson's theorem from approximation theory. The above five theorems in this paper are presented in \S 3, 4, 5, 6 and 7 in order.