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Statement

For vectors $\mathbf a,\mathbf b\in\mathbb R^n$ write $d(\mathbf a,\mathbf b)=(|a_1-b_1|,\dots,|a_n-b_n|)$, the coordinatewise absolute difference.

Theorem 4 (p. 3). If $A$ is a finite set of distinct vectors in $\mathbb R^n$, then $D(A)=\{d(\mathbf a,\mathbf b):\mathbf a,\mathbf b\in A\}$ contains at least $|A|$ distinct vectors.

The paper notes that $d(\mathbf a,\mathbf b)=\delta(\mathbf a,\mathbf b)+\delta(\mathbf b,\mathbf a)$ with $\delta(\mathbf a,\mathbf b)=(\max\{0,a_i-b_i\})_i$, so that for exponent vectors of integers $d(\mathbf a,\mathbf b)$ is the vector of $ab/\gcd(a,b)^2$, and that Theorem 4 is equivalent to Theorem 3 (p. 3).

Equality (section 3, p. 6). The paper says equality holds when $A$ is a suitable translate of a set $R\cap\Lambda$, with $R$ a box in $\mathbb Z^k$ with sides parallel to the axes and $\Lambda$ a lattice with $(2\mathbb Z)^k\subseteq\Lambda\subseteq\mathbb Z^k$, and only then; its Proposition 1 states the converse direction precisely ($|D(A)|=|A|$ forces $A=\{\mathbf a+\sum_j i_j\mathbf v_j:(i_1,\dots,i_k)\in I\}$ with $I$ of that form and each coordinate of $\mathbb R^n$ non-zero in at most one $\mathbf v_j$), with a proof the paper calls a sketch (pp. 6--7). In one dimension the two-set form, Proposition 2 (p. 6), gives $\#\{|a-b|:a\in A,b\in B\}\ge\min\{|A|,|B|\}$ for sets of distinct reals; the paper's section 4 (p. 7) asks whether $|D(A,B)|\ge\min\{|A|,|B|\}$ holds for finite sets of distinct vectors in $\mathbb R^n$, and leaves it open.

Source. A. Granville and F. Roesler, The set of differences of a given set, Amer. Math. Monthly 106 (1999), no. 4, 338--344; Theorem 4 on p. 3 of the authors' eight-page preprint, its proof on pp. 4--5, section 3 on pp. 6--7 and section 4 on p. 7, read on the page images. The journal version was not compared.

Read depth. Claims checked: the statement was read clause by clause on the page image of p. 3, and the proof (pp. 4--5) was followed step by step; section 3 was read for its statements only.

Proof pointer

Induction on $n$ and then on $|A|$ (pp. 4--5). One point or $n=1$ is direct. Otherwise let $B$ be the projection of $A$ to the first $n-1$ coordinates, and let $C$ be $A$ with the highest point above each $\mathbf b\in B$ removed, so $|C|=|A|-|B|$. Grouping $D(A)$ and $D(C)$ by the first $n-1$ coordinates, each value in $D(B)$ loses at least its largest last-coordinate difference in passing from $A$ to $C$, since that difference always involves a removed point; hence $|D(C)|\le|D(A)|-|D(B)|$, and the induction hypotheses for $B$ and $C$ give $|D(A)|\ge|B|+|C|=|A|$.

Dependencies

None outside the paper.

Bears on

None recorded. Through Theorem 3 it concerns the symmetric analogue of the ratio count of Problem 539; the paper derives no bound on that problem's $h(n)$ from it.