Let $X$, $T$, and $x_0$ be positive real numbers. Consider the region in $\mathbb{R}^2$ defined by $$ xy \leq X, \ \ x_0 \leq x \leq x_0 + T, \ \ \frac{X}{x_0 + T} \leq y. $$ Let $A$ be the area of this region and $N$ be the number of integers points in this region. Does it then follow that there exists some absolute constant $C$ such that $$ N \leq C \cdot A. $$ I would greatly appreciate any comments, suggestions, or references. Thank you very much! (There is also a possibility that I am missing something very simple here...)
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2$\begingroup$ I think you need to be more careful: it's possible to arrange the region to have exactly one point and arbitrarily small area. You can certainly get some bound like that with an additional O(T) additive error just by counting in columns. The standard method for counting points under a hyperbola with better error terms is Dirichlet's hyperbola method. $\endgroup$– Xiaoyu HeCommented Apr 19, 2017 at 0:31
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$\begingroup$ From Pick's theorem it follows that $$N\le 2A'+2,$$ where $A'$ is the area of the integer points convex hull. en.wikipedia.org/wiki/Pick%27s_theorem $\endgroup$– Max AlekseyevCommented Apr 19, 2017 at 10:38
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