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Number theory is a prominent branch of mathematics dedicated to the study of the integers. Within this is the study of special sets of numbers, such as the prime numbers, the squares, the squarefree numbers and many more. For such a set, we can ask two important questions:

• How many are there?
• How are they distributed?

In this thesis, higher dimensional analogues of these questions are studied. Rather than studying sets of numbers, we consider a broad class of sets consisting of tuples (a_1,…,a_n) of integers. For example, we study the set of triples (a_1,a_2,a_3) such that the product a_1 a_2 a_3 is squareful: every prime number dividing the product divides it at least twice.

This thesis introduces the geometric framework of so-called M-points, which allows the study of such sets using the tools and language from algebraic geometry. This framework is then used to give a broad generalisation of Manin’s conjecture, from rational points on varieties to M-points. In particular, this conjecture gives an asymptotic for the number of tuples (a_1,…,a_n) contained in the set with all coordinates less than B, as B goes to infinity.

Besides this, the thesis studies the distribution of such sets. This concerns whether the analogue of the Chinese remainder theorem holds for this set of M-points: when can p-adic M-points be lifted to M-points (over the integers)? In this thesis, this question is precisely answered for split toric varieties, extending known results on strong approximation for toric varieties.