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Statement

Theorem 2. For each $m\ge1$, a set $A$ of $m$ distinct vectors for which $|\delta(A)|$ is minimal satisfies

$$(3/2)(2m)^{2/3}\ \gtrsim\ |\delta(A)|\ \ge\ m^{1/2}.$$

Here $\delta(A)=\{\delta(\mathbf a,\mathbf b):\mathbf a,\mathbf b\in A\}$ with $\delta(\mathbf a,\mathbf b)=(\max\{0,a_i-b_i\})_i$, and "minimal" means minimal over all $m$-sets of distinct vectors (of any dimension), so $|\delta(A)|$ is the least value in the vector form of the paper's Unsolved problem, the $h(m)$ of Problem 539 (the paper gives it no symbol). The paper presents the theorem as "the following partial result concerning our unsolved problem", obtained "from these remarks, combined with those directly above the statement of Theorem 1" (p. 3): the upper estimate is the size of $\delta(A)$ for the Freiman–Lev sets in the plane and the lower estimate is the pairing argument of p. 2.

Source. A. Granville and F. Roesler, The set of differences of a given set, Amer. Math. Monthly 106 (1999), no. 4, 338--344; Theorem 2 on p. 3 of the author preprint, read on the page image; its two inputs on pp. 2--3. The journal version was not compared.

Read depth. Claims checked: the statement and the sentence introducing it were read clause by clause on the page image; the lower bound's argument was followed on p. 2 and is complete as printed; the count for the Freiman–Lev sets is the paper's and was not redone.

Proof pointer

Combination of the lower-bound paragraph on p. 2 (recorded on the Unsolved problem page) with the Freiman–Lev sets of pp. 2--3 (recorded on the Theorem 1 page).

Dependencies

The size count for the Freiman–Lev sets (the paper's, credited to Freiman and Lev without a reference).

Bears on

• Problem 539: in the site's notation $m^{1/2}\le h(m)\lesssim(3/2)(2m)^{2/3}$, the two refereed bounds the site's commentary credits to Erdős–Szemerédi and to Freiman–Lev, saying that a proof of both can be found in this paper. The 2026 upper bound $e^{O(\sqrt{\log m})}m^{1/2}$ attributed on the site to an automated proof system replaces the exponent $2/3$ by $1/2+o(1)$ through sets in growing dimension.