Petrov 2021 remark sets few distances

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theorem_1_1: Proves the Bannai--Bannai--Stanton upper bound for finite Euclidean s-distance sets using the inertia estimate in Theorem 1.2.

theorem_1_2: Bounds the rank and real inertia of a polynomial matrix by the dimension of low-degree polynomial functions on the indexing set.

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Petrov, Fedor and Pohoata, Cosmin, A remark on sets with few distances in $\mathbb{R}^{d}$. Proc. Amer. Math. Soc. 149 (2021), 569--571, DOI 10.1090/proc/15231. The copy read for this card is the three-page arXiv:1912.08181v1 (17 December 2019), the only arXiv version listed by the archive; its printed pages are the source pages used below. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1912.08181), every other right reserved.

Petrov and Pohoata give a new, simple proof of the Bannai--Bannai--Stanton theorem (their Theorem 1.1): for a positive integer $s$, any $s$-distance set $A$ in $\mathbb{R}^{d}$ satisfies $|A|\leq\binom{d+s}{s}$. The proof rests on Theorem 1.2, a strengthened real version of the Croot--Lev--Pach lemma: for a finite set $A$ in a finite-dimensional vector space $V$ over a field $F$ and a polynomial $p(\mathbf{x},\mathbf{y})$ in $2\dim V$ variables of degree at most $2s+1$, the matrix $M_{p,A}$ with entries $p(a,b)$ satisfies $\operatorname{rank}(p,A)\leq2\dim_s(A)$, and over the reals the inertia indices satisfy $\max\{r_+(p,A),r_-(p,A)\}\leq\dim_s(A)$, where $\dim_s(A)$ is the dimension of the space of degree-at-most-$s$ polynomials restricted to $A$. The mechanism is Sylvester's Law of Inertia for quadratic forms applied to the bilinear form $\Phi_p$, combined with the polynomial-method dimension count; only part 2 of Theorem 1.2 is needed to derive Theorem 1.1. For $s=2$ the bound is the upper bound in Erdős Problem 502, which concerns the largest size of a set in $\mathbb{R}^{d}$ determining exactly two distinct distances.

Source: https://arxiv.org/abs/1912.08181.

Bears on. #502: Theorem 1.1 with $s=2$ 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 bound is Bannai, Bannai and Stanton's (1983) and this paper gives a new proof of it, through part 2 of Theorem 1.2. The paper gives no lower bound and does not determine the exact maximum.

Read against the print. The statements of Theorems 1.1 and 1.2 and the definition of an $s$-distance set were checked clause by clause against the three printed pages; the proofs were read for structure. Two misprints are noted on the result pages: the definition of an $s$-distance set (p. 1) speaks of the distances "determined by the points in $M$" [sic] where the points of $A$ are meant, and a sum in the proof of Theorem 1.2 (p. 2) is indexed by "$b\in B$" [sic] where $b\in A$ is meant.

Results.

• Theorem 1.1 (p. 1): if $A$ is an $s$-distance subset of $\mathbb{R}^{d}$, then $|A|\leq\binom{d+s}{s}$ (the Bannai--Bannai--Stanton bound); deduced from Theorem 1.2 on p. 3.
• Theorem 1.2 (p. 2): for a nonnegative integer $s$, $p$ of degree at most $2s+1$ in $2\dim V$ variables over a field $F$ and finite $A\subset V$, $\operatorname{rank}(p,A)\leq2\dim_s(A)$; if $F=\mathbb{R}$, $\max\{r_+(p,A),r_-(p,A)\}\leq\dim_s(A)$. Proved on pp. 2--3.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.