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Statement

An $s$-distance set is a set of points in which the distance between distinct points takes only $s$ different values (the tract's usage, p. 1). $E^d$ is $\mathbb{R}^d$ with the usual inner product and metric; $H^d$ is $d$-dimensional hyperbolic space, realized on p. 26 as the lines $\langle x\rangle$ of $\mathbb{R}^{1,d}$ with $\langle x,x\rangle>0$.

Theorem 4.1.1 (p. 26). If $X$ is an $s$-distance set in $E^d$ or in $H^d$, then

$$\operatorname{card}(X)\le\binom{d+s}{s}.$$

The tract states the theorem in its introduction to Chapter 4 and proves the two cases separately: the Euclidean case is Theorem 4.3.1 (stated p. 27, proof pp. 28--30) and the hyperbolic case Theorem 4.4.1 (stated p. 30, proof following it). The introduction (p. 26) records that Koornwinder's argument gives the weaker bound $\binom{d+s}{s}+\binom{d+s-1}{s-1}$ in both spaces.

For $s=2$ the theorem says that a two-distance set in $\mathbb{R}^d$ has at most $\binom{d+2}{2}=\tfrac12(d+1)(d+2)$ points.

Source. A. Blokhuis, Few-distance sets, CWI Tract 7, Centrum voor Wiskunde en Informatica, Amsterdam, 1984; Theorem 4.1.1 on printed p. 26, Theorem 4.3.1 on p. 27 with Lemma 4.3.2 on p. 29, Theorem 4.4.1 on p. 30. The edition read is identified in the source digest.

Read depth. Claims checked: Theorems 4.1.1, 4.3.1 and 4.4.1 were read clause by clause on the page images. The proof of Theorem 4.3.1 was read for its structure, as sketched below; its computations were not checked, and the proof of Theorem 4.4.1 was not read. Nothing here is independently reviewed.

Proof pointer

Euclidean case (pp. 28--30). Let $\alpha_1,\ldots,\alpha_s$ be the squared distances occurring in $X$, and attach to each $u\in X$ the polynomial $F_u(x)=\prod_i(|x-u|^2-\alpha_i)$, which vanishes at every point of $X$ but $u$, so the $F_u$ are linearly independent. Each $F_u$ is a combination of the functions $(x,x)^{\delta}x^{b}$ with $\delta+\beta=s$, or with $\delta=0$ and $\beta<s$ (where $\beta$ is the degree of the monomial $x^b$), a space of dimension $\binom{d+s}{s}+\binom{d+s-1}{s-1}$; this alone is Koornwinder's bound. The improvement shows that the $F_u$ together with all monomials $x^b$ of degree $\beta<s$ are still independent: in a dependency relation, Lemma 4.3.2 (p. 29) shows, by induction on the degree and a sum-of-squares argument on the homogeneous parts, that $\sum_u a_uu^b=0$ for every $b$ with $\beta<s$, and evaluating the relation at each $u\in X$ then forces every $a_u=0$. Counting dimensions gives $\operatorname{card}(X)+\binom{d+s-1}{s-1}\le\binom{d+s}{s}+\binom{d+s-1}{s-1}$.

Dependencies

Within the tract: Lemma 4.3.2 (p. 29) for the Euclidean case and the model of $H^d$ of §4.2 (p. 26) for the hyperbolic case.

Bears on

• Problem 502: the case $s=2$ in $E^d$ bounds the size of a two-distance set in $\mathbb{R}^d$ by $\binom{d+2}{2}$, an upper bound on the quantity the problem asks for.
• Problem 503: the proof of Theorem 7.2.5 (p. 49) uses the case $s=2$ for an isosceles set that is a two-distance set.