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Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$. Say that a point $(x,y)$ of $Q$ is visible from the origin if the segment from $(0,0)$ to $(x,y) \in Q$ passes through no other point of $Q$. So points block visibility, and the only points visible from the origin are those with a clear line of sight:
   
Let $\nu(n)$ be the ratio of the number of points visible within the square with corner $(n,n)$ to $n^2$. For example, for $n=8$, $43$ of the $64$ points are visible, so $\nu(n) = 43/64 \approx 0.67$.

Q: What is $\lim_{n \to \infty} \nu(n)$?

It appears to be approaching $\approx 0.614$:
   
The question bears some similarity to Polya's Orchard Problem (T.T. Allen, "Polya's orchard problem," The American Mathematical Monthly 93(2): 98-104 (1986). Jstor link), but I cannot see my question is answered in that literature.

One could ask the same question for $\mathbb{Z}^d$.

Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$. Say that a point $(x,y)$ of $Q$ is visible from the origin if the segment from $(0,0)$ to $(x,y) \in Q$ passes through no other point of $Q$. So points block visibility, and the only points visible from the origin are those with a clear line of sight:
   
Let $\nu(n)$ be the ratio of the number of points visible within the square with corner $(n,n)$ to $n^2$. For example, for $n=8$, $43$ of the $64$ points are visible, so $\nu(n) = 43/64 \approx 0.67$.

Q: What is $\lim_{n \to \infty} \nu(n)$?

It appears to be approaching $\approx 0.614$:
   
The question bears some similarity to Polya's Orchard Problem (T.T. Allen, "Polya's orchard problem," The American Mathematical Monthly 93(2): 98-104 (1986). Jstor link), but I cannot see my question is answered in that literature.

One could ask the same question for $\mathbb{Z}^d$.

Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$. Say that a point $(x,y)$ of $Q$ is visible from the origin if the segment from $(0,0)$ to $(x,y) \in Q$ passes through no other point of $Q$. So points block visibility, and the only points visible from the origin are those with a clear line of sight:
   
Let $\nu(n)$ be the ratio of the number of points visible within the square with corner $(n,n)$ to $n^2$. For example, for $n=8$, $43$ of the $64$ points are visible, so $\nu(n) = 43/64 \approx 0.67$.

Q: What is $\lim_{n \to \infty} \nu(n)$?

It appears to be approaching $\approx 0.614$:
   
The question bears some similarity to Polya's Orchard Problem (T.T. Allen, "Polya's orchard problem," The American Mathematical Monthly 93(2): 98-104 (1986). Jstor link), but I cannot see my question is answered in that literature.

Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$. Say that a point $(x,y)$ of $Q$ is visible from the origin if the segment from $(0,0)$ to $(x,y) \in Q$ passes through no other point of $Q$. So points block visibility, and the only points visible from the origin are those with a clear line of sight:
   
Let $\nu(n)$ be the ratio of the number of points visible within the square with corner $(n,n)$ to $n^2$. For example, for $n=8$, $43$ of the $64$ points are visible, so $\nu(n) = 43/64 \approx 0.67$.

Q: What is $\lim_{n \to \infty} \nu(n)$?

It appears to be approaching $\approx 0.614$:
   
The question bears some similarity to Polya's Orchard Problem (T.T. Allen, "Polya's orchard problem," The American Mathematical Monthly 93(2): 98-104 (1986). Jstor link), but I cannot see my question is answered in that literature.

One could ask the same question for $\mathbb{Z}^d$.