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Statement

For a positive integer $s$, the paper (p. 1) calls a finite subset $A$ of a metric space an $s$-distance set when there are $s$ positive reals $d_1,\dots,d_s$ such that every distance between two distinct points of $A$ is one of them and each $d_i$ occurs. (The printed definition says the distances "determined by the points in $M$" [sic], the ambient space; the points of $A$ are meant.) In $\mathbb{R}^d$ with the Euclidean distance:

Theorem 1.1 (p. 1). "If $A$ is an $s$-distance subset in $\mathbb{R}^d$, then $|A|\leq\binom{d+s}{s}$."

The paper attributes the theorem to Bannai, Bannai and Stanton (1983), its reference [1]; what is new in the note is the proof. The deduction on p. 3 ends with the same bound written as $\binom{s+d}{d}$, which is equal.

Source. Fedor Petrov and Cosmin Pohoata, A remark on sets with few distances in $\mathbb{R}^{d}$, Proc. Amer. Math. Soc. 149 (2021), 569--571, read in the arXiv:1912.08181v1 edition identified on the source card: Theorem 1.1 stated on p. 1, deduced from Theorem 1.2 on p. 3.

The original source of the bound is E. Bannai, E. Bannai and D. Stanton, An upper bound for the cardinality of an $s$-distance subset in real Euclidean space, II, Combinatorica 3 (1983), 147--152, DOI 10.1007/BF02579288.

Read depth. Claims checked: the statement and the definition of an $s$-distance set were read clause by clause on the print; the deduction on p. 3 was read for structure.

Proof pointer

The deduction (p. 3) applies part 2 of Theorem 1.2 over $\mathbb{R}$ with $V=\mathbb{R}^d$. Take the polynomial in $2d$ variables that is the product, over the $s$ distances $\delta$ of $A$, of $\delta^2-\|\mathbf x-\mathbf y\|^2$; its degree is $2s\le 2s+1$. On $A\times A$ it vanishes off the diagonal and equals the same positive number on the diagonal, so its matrix is a positive multiple of the identity and the positive inertia index is $|A|$. Theorem 1.2 bounds this by $\dim_s(A)$, which is at most the dimension $\binom{d+s}{s}$ of all polynomials of degree at most $s$ on $\mathbb{R}^d$.

Dependencies

Theorem 1.2, part 2, and the count of monomials of degree at most $s$ in $d$ variables.

Bears on

• Problem 502: the case $s=2$ bounds every two-distance set in $\mathbb{R}^d$ by $\binom{d+2}{2}$ points, the upper bound on the largest two-distance set. The theorem is Bannai, Bannai and Stanton's; this paper gives a new proof of it. It gives no lower bound and does not determine the exact maximum.