I created a code which calculates
$$D(S\cap[a,b])=\lim_{n\to\infty}\frac{\left|S\cap{F_n\cap[a,b]}\right|}{\left|F_n\cap[a,b]\right|}$$
where $D$ is the density of $S\cap[a,b]$ (in $A\cap[a,b]$), $[a,b]$ is an interval for $a,b\in\mathbb{R}$, $F_n$ is the Følner Sequence of $A$, and $S\subseteq A$. For more information, click here (replace $G,X,i,g$ with $A,S,n,a$), click (here) and here.
I set $A=\left\{\frac{p}{2^k(2q+1)}:p,q,k\in\mathbb{Z},k\ge 0\right\}$, $F_n=\left\{\frac{p}{2^k(2q+1)}:p,q,k\in\mathbb{Z},k\ge 0,k\le n,|2q+1|\le n\right\}$ and $S=\left\{\frac{j^2+j+1}{k^3+1}:j,k\in\mathbb{Z},k\neq -1\right\}$
In my code $A$ is A[p_,k_,q_], $F_n$ is f[n_,a_,b_], $S$ is S[j_,k_] and $D$ is d.
Clear[A, F, f, p, Ff, S, X, Y, d, j, k];
A[p_, k_, q_] := p/((2^k)*(2*q + 1));
F[p_, n_] :=
Table[A[p, k, q], {k, 0, Floor[Log[2, n]]}, {q, 0,
Floor[(n - 1)/2]}];
f[n_, a_, b_] :=
p /. Table[
Solve[a <= A[p, k, q] <= b, p, Integers], {k, 0,
Floor[Log[2, n]]}, {q, 0, Floor[(n - 1)/2]}];
Ff[n_, a_, b_] :=
DeleteDuplicates@
Flatten@Table[
F[f[n, a, b][[v]][[u]], n][[v]][[u]], {v, 1,
Floor[Log[2, n]] + 1}, {u, 1, Floor[(n - 1)/2] + 1}];
S[j_, k_] := (j^2 + j + 1)/(k^3 + 1);
X[n_, a_, b_] :=
Count[Resolve[
Exists[{j, k}, S[j, k] == # && {j, k} \[Element] Integers]] & /@
Ff[n, a, b], True];
Y[n_, a_, b_] := Length[Ff[n, a, b]];
d[n_, a_, b_] := N[(X[n, a, b])/Y[n, a, b]];
Ff[4, 1, 2]
Num = X[10, 1, 2]
Den = Y[10, 1, 2]
Timing[N[Num/Den]]
My result is
{1, 2, 4/3, 5/3, 3/2, 7/6, 11/6, 5/4, 7/4, 13/12, 17/12, 19/12, 23/12}
10
153
{0., 0.0653595}
Everything seems correct (Edit: I was wrong it is not correct) except the timing since the calculation must have taken more than 5 minutes.
I want to calculate n at higher numbers as quickly as possible. In fact, I want to calculate Limit[d[n,a,b],n->Infinity].
I was suggested in this post to use methods similar to monte Carlo integration? Are my new methods correct?
Edit: I found out Exists is incorrectly calculating X[n,a,b] for large n
Example: If we set $S=\left\{\frac{m^2}{n^3}:m,n\in\mathbb{Z},n\neq 0\right\}$, I get N[Num/Den] is approximately zero instead of $1$. This makes no sense since $\left\{\frac{m^2}{n^3}:m,n\in\mathbb{Z},n\neq 0\right\}=\left\{\frac{a}{b}:a,b\in\mathbb{Z},b\neq 0\right\}$, so the density should be $1$.
How do we correct Exists?
