The proposal is concerned with applying classical and more recent forms of the combinatorial sieve, in the setting of an orbit of a group of morphisms of affine n-space,which preserves integer points. The setting unifies and generalizes the problem of finding points at which a polynomial takes on values which are prime or has few prime factors.This "affine sieve" has numerous applications to both classical and novel diophantine problems. The methods used to develop an effective sieve in this context involve automorphic forms, expander graphs and unexpectedly arithmetic and additive combinatorics.
The Twin Prime Conjecture asserts that there are infinitely many pairs of prime numbers which differ by two. It is one of the longest standing unsolved problems in mathematics. While such problems are driven first by curiosity, the techniques that have been invented for their study have proven to be fundamental more broadly. The proposal is concerned with far-reaching generalizations of the twin prime conjecture and with developing new techniques to prove parts of these general conjectures. The interplay between number theory, combinatorics and theoretical computer science has been a very active one in recent years. Many times in this context the applications have been of number theoretic ideas to the other two fields. In the present project the reverse application is also critical.
" was the development of tools to attack diophantineproblems connected with orbits of integral affine linear actions. The genesis of these problems are classical problems of representationsof integers by integral quadratic forms and their generalizations to arithmetic groups and automorphic forms. However there are manyproblems of this type where the group in question is not arithmetic (being of infinite index in the natural group of integer points inwhich they lie and which we call "thin matrix groups") and which arise naturally. For example in integral apollonian packingsand in monodromy groups associated with integral cohomolgy groups of families of varieties. Basic techniques to tackle these diophantineproblems ranging from the execution of a fundamental sieve (a-la Brun) to the study of local to global principles in these settings,are now in place. These have been achieved in part by the PI, his collaborators and students in connection with this grant. Perhaps themost novel aspects of these tools come from the application of notion of expander graphs from computer science and the establishment of thekey expander property in the context of these affine actions. These results which are a direct outgrowth of this grant have been exposedin many surveys, conferences and public lectures. We list below three surveys that give an account of some of these developments. Variousother related technical results that were established for these applications as well as for their independent interest, are listed inthe annual reports for the grant. Kontorovich, Alex From Apollonius to Zaremba: local-global phenomena in thin orbits. Bull. Amer. Math. Soc. (N.S.)50 (2013), no. 2, 187–228 Fuchs, Elena Counting problems in Apollonian packings. Bull. Amer. Math. Soc. (N.S.) 50 (2013), no. 2, 229–266. (P.Sarnak )"Notes on thin matrix groups", in "Thin Groups and Superstrong Approximations" 343-362,MSRI Publications Vol 61 (2014), Editors E. Breuillard and H. Oh
