Herzog 1971 patterns visible nonvisible lattice points

../

corollary_2: Herzog and Stewart's corollary that every planar pattern with one, two or three prescribed visible points and any number of prescribed nonvisible points is realizable, giving visible points isolated from all others by any distance; the paper uses it to show the visible points are not connected.

theorem_1: Herzog and Stewart's criterion in the plane: a pattern prescribing visible and nonvisible points on a square block occurs as a translate in the integer lattice exactly when its prescribed visible points contain no complete residue square modulo p for any prime p, whatever the nonvisible points.

theorem_2: Herzog and Stewart's extension of their planar criterion to dimension k >= 3: a pattern of visible and nonvisible points is realizable in the k-dimensional integer lattice exactly when its prescribed visible points contain no complete residue hypercube modulo p for any prime p.

---

Fritz Herzog, B. M. Stewart, Patterns of Visible and Nonvisible Lattice Points. The American Mathematical Monthly 78(5) (1971), 487-496. doi:10.2307/2317753. The copy read for this card is the JSTOR PDF. Its cover sheet prints "Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms" and names the publisher as "Taylor & Francis on behalf of the Mathematical Association of America", and every page carries "All use subject to https://about.jstor.org/terms", the platform's notice, recorded as reserved. Crossref lists a correction to the paper in the same volume, p. 870 (doi:10.2307/2316477); it was not read for this card.

A pattern prescribes, on the points of a $w\times\cdots\times w$ block of $L_k$, which are to be visible (circles: coordinates with no common divisor greater than $1$) and which nonvisible (crosses); it is realized when some translate of the block meets the prescription (pp. 487--488). Theorem 1 (p. 490) states that a planar pattern is realizable if and only if, for every prime $p$, its set of circles contains no complete square modulo $p$ (a set of $p^2$ points meeting every residue class of $L_2$ modulo $p$ exactly once); realizability therefore depends only on the circles. Necessity holds because every translate of a complete square modulo $p$ again has a point $\equiv(0,0)$ modulo $p$; sufficiency is a three-step Chinese Remainder Theorem construction: congruences modulo the primes $p\le w$ keep the circles off $(0,0)$ modulo $p$, a separate prime $Q(i,j)>w$ for each cross makes it nonvisible, and a final step with $u$ fixed puts $v\equiv0$ modulo every remaining prime $q>w$ dividing one of $u+1,\ldots,u+w$ (pp. 490--492). Theorem 2 (p. 495) extends the criterion to $L_k$, $k\ge3$, with complete $k$-dimensional hypercubes modulo $p$; in the last step only the second coordinate $u_2$ needs the extra congruences, since a prime that does not divide both of the first two coordinates of a point cannot divide all of them.

Section 3 (pp. 492--495) lists corollaries with numerical examples: Corollary 1, every planar pattern consisting only of crosses is realizable; Corollary 2, every pattern with one, two or three circles and any number of crosses is realizable, so there are visible points separated from all other visible points by an arbitrarily great distance; Corollary 3, the rectangle with vertices $(1,1),(M,1),(M,N),(1,N)$, $M\ge2$, $N\ge2$, with circles on its boundary and crosses inside, is realizable if and only if $M$ and $N$ are both odd; Corollary 4, the rectangle with vertices $(\pm m,\pm n)$, $m\ge1$, $n\ge1$, with circles at the origin and on the boundary and crosses at the other interior points, is realizable if and only if $6\mid mn$; Corollary 5, the square diamond with vertices $(\pm m,0)$, $(0,\pm m)$, $m\ge1$, with circles on its edges and crosses inside, is realizable for all $m$.

The introduction (p. 489) draws from these theorems (see also Corollary 1) that $L_k$ contains arbitrarily large hypercubes of nonvisible points, although the visible points have relative frequency $1/\zeta(k)$, and proves that density for $k\ge3$ (pp. 489--490). On p. 490 the paper calls a set connected when any two of its points are joined by a chain of points of the set at successive distance $1$, and states, by Corollary 2 for $k=2$ and its analogue for $k>2$, that the set $V_k$ of visible points is not connected. It announces a subsequent paper on the connected components of the visible and of the nonvisible points, particularly in $L_2$. The paper does not consider paths to infinity.

Source: https://www.jstor.org/stable/2317753.

Read status: claims checked for the definitions, Theorems 1 and 2, Corollaries 1 to 5, the density argument and the connectedness paragraph, read clause by clause on the page images of the print; the proofs of Theorems 1 and 2 followed. Nothing here is independently reviewed. Result pages: theorem_1, theorem_2 and corollary_2.

Bears on. #1212: the paper's connectedness uses unit steps between visible points of $L_2$, the problem's adjacency taken over all of $L_2$, and Corollary 2 (p. 493) yields its statement that the visible points are not connected (p. 490). The paper does not consider paths to infinity, the condition $\min(x,y)>1$ or composite coordinates, and does not address the problem's question.

Results.

• Theorem 1 (p. 490), with Corollary 1 (p. 492): a planar pattern is realizable in $L_2$ if and only if its circles contain no complete square modulo $p$ for any prime $p$.
• Theorem 2 (p. 495): for $k\ge3$, a pattern is realizable in $L_k$ if and only if its circles contain no complete $k$-dimensional hypercube modulo $p$ for any prime $p$.
• Corollary 2 (p. 493): every planar pattern with one, two or three circles is realizable, giving arbitrarily lonesome visible points; with the connectedness paragraph of p. 490.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.