In the paper under review, the author is concerned with integral distances from given lattice points, which here mean points in the integral lattice \( \mathbb{Z}^2 \), that is, points in the plane \( \mathbb{R}^2 \) having integer coordinates.
Here, \( (\cdot, \cdot)\) denotes the usual scalar product in \( \mathbb{R}^2 \), and \( |\cdot| \) denotes the associated distance, so that \( |Q-P| \) is the (usual Euclidean) distance between two points \( P \) and \( Q \). Also, for \( P_1, P_2, \ldots, P_r \in \mathbb{Z}^2 \) distinct lattice points, set \begin{align*} \mathscr{S}=\mathscr{S}(P_1, P_2, \ldots, P_r)=\left\{Q\in \mathbb{Z}^2: ~|Q-P_i|\in \mathbb{Z} ~\text{ for }~i=1,2,\ldots, r\right\}. \end{align*} The author is mainly interested in understanding when the set \( \mathcal{S} \) is infinite, and possibly in describing its distribution. For \( r=1 \), it is just the Pythagorean triple \((a,b,c)\), completely parametrized by the formula \( a^2+b^2=c^2 \). When \( r=2 \) and the distance \( |P_1-P_2| \) is integral, it is easy to see that for points \( Q \), the triangle, \( P_1P_2Q \) is Heronian, that is, has integral sides and integral area. For \( r\ge 3 \), the set \( \mathcal{S} \) is finite and effectively computable unless all the \( P_i \) are collinear and have mutual integral distances. The infinitude of the set \( \mathcal{S} \) occurs when \( r=2 \), the case which is the motivation for this paper. The main result of this paper is the following.
Theorem 1. For every finite union \( \mathscr{L} \) of lines, the set \( \mathscr{S}(P_1,P_2) \backslash \mathscr{L} \) is infinite, unless the point \( P:=P_2-P_1 \), after possible sign changes and switching of its coordinates, belongs to the following list (where we replace by translation \( P_1, P_2 \), respectively, by \( O , P=P_2-P_1\):

(i)

\( P=(0,1) \): now \( \mathscr{S} \) consists of integer points on the \( y \)-axis.

(ii)

\( P=(1,1) \): now \( \mathscr{S} \) is infinite and contained in the line \( x+y=1 \).

(iii)

\( P=(0,2) \): now \( \mathscr{S} \) consists of integer points on the \( y \)-axis.

(iv)

\( P=(1,1) \): now \( \mathscr{S} =\{(1,0), (0,2)\}\).

(v)

\( P=(2,2) \): now \( \mathscr{S} \) is infinite and contained in the line \( x+y=2 \).

(vi)

\( P=(1,1) \): now \( \mathscr{S} \) is the union of an infinite set contained in the line \( 2x+y=5 \) and the set \( \{(0,2), (4,0), (-1,0), (3,4)\} \).

The proof of Theorem 1 uses only elementary arguments, the crucial ingredient being a theorem of Gauss which does not appear to be often invoked. Furthermore, the author includes related remarks and open questions for integral and rational distances from an arbitrary prescribed finite set of lattice points.