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Register a new userA Conjectural Inequality for Visible Points in Lattice Parallelograms
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Let $a,n in mathbb{Z}^+$, with $a<n$ and $gcd(a,n)=1$. Let $P_{a,n}$ denote the lattice parallelogram spanned by $(1,0)$ and $(a,n)$, that is, $$P_{a,n} = left{ t_1(1,0)+ t_2(a,n) , : , 0leq t_1,t_2 leq 1 right}, $$ and let $$V(a,n) = # textrm{ of visible lattice points in the interior of } P_{a,n}.$$ In this paper we prove some elementary (and straightforward) results for $V(a,n)$. The most interesting aspects of the paper are in Section 5 where we discuss some numerics and display some graphs of $V(a,n)/n$. (These graphs resemble an integral sign that has been rotated counter-clockwise by $90^circ$.) The numerics and graphs suggest the conjecture that for $a ot= 1, n-1$, $V(a,n)/n$ satisfies the inequality $$ 0.5 < V(a,n)/n< 0.75.$$
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This paper concerns the number of lattice points in the plane which are visible along certain curves to all elements in some set S of lattice points simultaneously. By proposing the concept of level of visibility, we are able to analyze more carefully about both the visible points and the invisible points in the definition of previous research. We prove asymptotic formulas for the number of lattice points in different levels of visibility.
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