Manin’s conjecture for Fano varieties predicts an asymptotic formula for the number of rational points of bounded height with respect to the anti-canonical height function on a small enough Zariski open set with a dense set of rational points. In the case of toric varieties, Manin’s conjecture was verified by Victor Batyrev and Yuri Tschinkel. Emmanuel Peyre has proposed two notions, (i) freeness and (ii)all the heights approach to delete accumulating subvarieties in (i) Liberte et accumulation and (ii) Beyond heights: slopes and distribution of rational points.

In this talk, based on the all the heights approach, I will explain a multi-height variant of the Batyrev-Tschinkel theorem where one considers working at {\em height boxes}, instead of a single height function, as a way to get rid of accumulating subvarieties. The main result of my joint work with Ramin Takloo-Bighash is the following: Let \(X\) be an arbitrary toric variety over a number field \(F\), and let \(H_i\), \(1 \leq i \leq r\), be height functions associated to the generators of the cone of effective divisors of \(X\). Fix positive real numbers \(a_i\), \(1 \leq i \leq r\). Then the number of rational points \(P \in X(F)\) such that for each \(i\), \(H_i(P) \leq B^{a_i}\) as \(B\) gets large is equal to \(C B^{a_1 + \dots + a_r} + O(B^{a_1 + \dots + a_r-\epsilon})\) for an \(\epsilon >0\). This result is a first example of a large family of varieties along the lines of Peyre’s idea.