Suppose, $1 < a_1\leq a_2\leq….\leq a_n$ are n-real numbers. Consider the hyperplanes defined by the equations,
$x_1=0,\;x_2=0,…….,x_n=0$ and $\frac{x_1}{a_1}+\frac{x_2}{a_2}+…+\frac{x_n}{a_n}=1$….. (1)
I want to know a sharp upper bound on the number of lattice points inside (1). That is, the cardinality of the set of points $(x_1, x_2,…,x_n)\in (\mathbb{Z}^+)^n$ such that $\frac{x_1}{a_1}+\frac{x_2}{a_2}+…+\frac{x_n}{a_n} < 1.$ Indeed I will be thankful if somebody could give me some references.
Thanks.
Best Answer
Here are some further thoughts in addition to my comment. Suppose that $a_1,\dots,a_n$ are integers $>1$. Write $\alpha=(a_1,\dots,a_n)$ and let $N(\alpha)$ be the number of integer vectors $(x_1,\dots,x_n)$ satisfying $x_i> 0$ and $\frac{x_1}{a_1}+\cdots+\frac{x_n}{a_n}< 1$. Let $\mathrm{lcm}(a_1,\dots,a_n)$ denote the least common multiple of $a_1,\dots,a_n$. Let $u$ be a positive integer. If my computations are correct, the Ehrhart theory gives that as $u\to\infty$, $$ N(u\alpha) = \frac{a_1\cdots a_n}{n!}u^n - \frac{a_1\cdots a_n}{2(n-1)!}\left( \frac{1}{a_1} +\cdots+\frac{1}{a_n} +\frac{1}{\mathrm{lcm}(a_1,\dots,a_n)}\right)u^{n-1} +O(u^{n-2}). $$ It seems reasonable that the above formula with $u=1$ will be a good approximation to $N(\alpha)$ when all $a_i$'s are large (and integers). It should be a better approximation than the first term, which just comes from the volume. (The second term comes from the "relative surface area.")
If $\epsilon>0$ is sufficiently small, then the points counted by $N((u+\epsilon)\alpha)$ will be the same as those counted by $N(u\alpha)$, except for the additional points satisfying $x_i>0$ and $\frac{x_1}{a_1}+\cdots+\frac{x_n}{a_n}=1$. We then get $$ N((u+\epsilon)\alpha) = \frac{a_1\cdots a_n}{n!}u^n - \frac{a_1\cdots a_n}{2(n-1)!}\left( \frac{1}{a_1} +\cdots+\frac{1}{a_n} -\frac{1}{\mathrm{lcm}(a_1,\dots,a_n)}\right)u^{n-1} +O(u^{n-2}). $$ This suggests the following question. Suppose that the $a_i$'s are any real numbers $>1$, as in the statement of the problem. What are the lim sup and lim inf of $$ u^{-n+1}\left(N(u\alpha)-\frac{a_1\cdots a_n}{n!}u^n\right) $$ as $u\to\infty$, $u\in\mathbb{R}$?