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Statement

Let $V$ be a finite-dimensional vector space over a field $F$, let $A\subset V$ be finite, and let $s$ be a nonnegative integer. Let $p(\mathbf x,\mathbf y)$ be a polynomial in $2\cdot\dim V$ variables with coefficients in $F$ and degree at most $2s+1$. Let $M_{p,A}$ be the matrix with rows and columns indexed by $A$ whose $(a,b)$ entry is $p(a,b)$; it is the matrix of the bilinear form on $F^A$

$$\Phi_p(f,g)=\sum_{a,b\in A}p(a,b)f(a)g(b)\qquad(f,g:A\to F),$$

which need not be symmetric, and $\Phi_p(f,f)$ is the associated quadratic form. Write $\operatorname{rank}(p,A)$ for the rank of $M_{p,A}$; when $F=\mathbb{R}$, write $r_+(p,A)$ and $r_-(p,A)$ for the positive and negative inertia indices of the quadratic form $\Phi_p(f,f)$ (the printed definition writes them $r_+(p)$, $r_-(p)$; the conclusion writes $r_\pm(p,A)$). Let $\dim_s(A)$ be the dimension of the space of polynomials of degree at most $s$ regarded as functions on $A$.

Theorem 1.2 (p. 2). Under these hypotheses:

1. "$\operatorname{rank}(p,A)\leqslant 2\dim_s(A)$."
2. "if $\mathbb{F}=\mathbb{R}$, then $\max\{r_+(p,A),r_-(p,A)\}\leqslant\dim_s(A)$."

The paper presents this as a slightly improved real version of the Croot--Lev--Pach lemma (its reference [4], Lemma 1); part 1 is, in its words (p. 2), "more or less the original Croot-Lev-Pach lemma in disguise", and only part 2 is used for Theorem 1.1.

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 1.2 stated on p. 2, proved in Section 2, pp. 2--3.

Read depth. Claims checked: the statement was read clause by clause on the print; the proof (pp. 2--3) was read for structure.

Proof pointer

Let $\Omega\subseteq F^A$ be the functions orthogonal, under the coordinate pairing $\sum_{a\in A}f(a)g(a)$, to every polynomial of degree at most $s$ restricted to $A$; its dimension is at least $|A|-\dim_s(A)$. Each monomial $\mathbf x^\alpha\mathbf y^\beta$ of $p$ has $|\alpha|\le s$ or $|\beta|\le s$, so the double sum it contributes factors into two single sums, one of which vanishes on $\Omega$; hence $\Phi_p$ is zero on $\Omega\times\Omega$. In a basis extending one of $\Omega$, the nonzero entries of the matrix lie in $|A|-\dim\Omega$ rows and as many columns, which gives part 1. Over $\mathbb{R}$, a subspace on which $\Phi_p(f,f)$ is positive definite meets $\Omega$ only in $0$, which bounds $r_+(p,A)$ by $|A|-\dim\Omega$; the same argument for $-\Phi_p$ bounds $r_-(p,A)$, giving part 2. The displayed factorization on p. 2 indexes its second sum by "$b\in B$" [sic]; $B$ is not defined, and $b\in A$ is meant.

Dependencies

Linear algebra only: the dimension of an annihilator under a nondegenerate pairing and Sylvester's law of inertia. No result of another paper is used.

Bears on

• Problem 502: part 2 is the lemma from which Theorem 1.1 derives the upper bound $\binom{d+2}{2}$ on two-distance sets in $\mathbb{R}^d$; on its own it bounds no distance set.