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Statement

Setting (p. 122). The paper turns to the problem of inequality (1) in $d$ dimensions, $d\ge3$: $k$ is the number of lattice points with all mutual distances distinct. The print does not restate the region; it is read here as the $d$-dimensional analogue of the plane's, integer coordinates in $(0,n]$.

Inequality (7) (p. 122). Replacing Landau's theorem by the theorems on sums of three or four squares, the paper gets

$$k<c_7\,d^{1/2}\,n,$$

with $c_7$ a positive constant; no range of $n$ is printed.

Conjecture (8) (p. 122). Marked with "(?)", the paper says the corresponding heuristic argument suggests

$$k<c_8\,d^{2/3}n^{2/3}(\log n)^{1/3}.$$

Lower bound (p. 122). The construction with (hyper)spheres and (hyper)planes gives the same lower bound (4), $k>n^{2/3-\varepsilon}$; no detail is given.

Proof pointer

p. 122, one sentence: the argument of inequality (2) with the theorems on sums of three or four squares in place of Landau's theorem. The squared distances are integers below $dn^2$, so $\binom k2<dn^2$, which gives $k<c_7d^{1/2}n$; the paper does not write out this count.

Read depth

Claims checked: (7), (8) and the remark on the lower bound were read clause by clause on the page image of p. 122. The count in the proof pointer is the corpus's reading of the paper's one-sentence derivation.

Dependencies

The theorems on sums of three or four squares, which the paper names but does not cite.

Source. P. Erdős, R. K. Guy, Distinct distances between lattice points, Elem. Math. 25 (1970), 121--123; the edition read is named on the source card.

Bears on

• Problem 1208: with the region read as above, the $N=n^d$ lattice points are one set of $N$ points in $\mathbb R^d$, so for each $n$ at which (7) holds, $F_d(n^d)<c_7d^{1/2}n$, that is $F_d(N)$ is $O(N^{1/d})$ along the $d$-th powers for fixed $d\ge3$. This is an upper bound only; the paper does not state it in terms of $F_d$ and gives no lower bound for $F_d$. Conjecture (8) concerns the lattice points, not arbitrary sets.