---

Statement

Setting (p. 495). Here $k\ge3$. Patterns $P_k$, circles, crosses and realization are as on the Theorem 1 page. A complete $k$-dimensional hypercube modulo $m$ is defined exactly as the complete square modulo $m$: a set of $m^k$ points of $L_k$ containing exactly one representative of each residue class of $L_k$ modulo $m$.

Theorem 2 (p. 495, quoted). "A given pattern $P_k$ can be realized in $L_k$ if and only if the set $C$ of circles in $P_k$ fails to contain a complete $k$-dimensional hypercube modulo $p$ for every prime $p$."

Consequence drawn in the introduction (p. 489). From the preceding remarks on Theorems 1 and 2 (see also Corollary 1), $L_k$ contains arbitrarily large hypercubes consisting entirely of nonvisible points, although the visible points of $L_k$ have relative frequency $1/\zeta(k)$. For $k\ge3$ the paper proves the density (pp. 489--490): with $\Psi_k(t)$ the number of visible points with $1\le x_\lambda\le t$, it shows $\Psi_k(t)=\sum_{d\ge1}\mu(d)[t/d]^k$ and that omitting the brackets costs at most $kt^{k-1}\zeta(k-1)=o(t^k)$, so $\Psi_k(t)/t^k\to1/\zeta(k)$ as $t\to+\infty$. For $k=2$ it cites Rademacher's book.

Proof pointer

P. 495. Necessity and the first two steps of sufficiency follow the proof of Theorem 1 with congruences (5') and (6'). In the third step $u_1$ is fixed positive and only $u_2$ receives the extra congruences (7'), $u_2\equiv0\bmod q$, because a prime that does not divide both of the first two coordinates of a point cannot divide all $k$ of them. Section 4 closes with a numerical realization of the $2\times2\times2$ cube of crosses.

Read depth

Claims checked: the definition, Theorem 2, its proof and the density argument of pp. 489--490 were read clause by clause on the page images of the print. Nothing here is independently reviewed.

Dependencies

None in the corpus.

Source. Fritz Herzog and B. M. Stewart, Patterns of visible and nonvisible lattice points, Amer. Math. Monthly 78 (1971), no. 5, 487--496, doi:10.2307/2317753; the edition read is named on the source card.

Bears on

No problem page of this corpus.