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- JEAN BOURGAIN, ALEX GAMBURD, PETER SARNAK
- 2008

- RAMANUJAN GRAPHS, GIULIANA DAVIDOFF, PETER SARNAK, ALAIN VALETTE
- 2003

Includes bibliographical references and index.

A basic problem of diophantine analysis is to investigate the asymptotics as T of (1.2) N(T, V)= {m V(Z): Ilmll T} where we denote by V(A), for any ring A, the set of A-points of V. Hence I1" is some Euclidean norm on R". The only general method available for such problems is the Hardy-Littlewood circle method, which however has certain limitations,… (More)

- Peter Sarnak
- 2003

These notes attempt to describe some aspects of the spectral theory of modular surfaces. They are by no means a complete survey.

- J. Bourgain, P. Sarnak, T. Ziegler
- 2011

We formulate and prove a finite version of Vinogradov's bilin-ear sum inequality.We use it together with Ratner's joinings theorems to prove that the Mobius function is disjoint from discrete horocycle flows on Γ\SL 2 (R).

1. Background. The aim of this note is to give an explicit construction of a rich family of k-regular (except for k ° =k) of the adjacency matrix satisfy Ikjl < 2 k~-l. graphs for which all the eigenvalues kj This bound is optimal (see Proposition 2.1). We call such graphs Ramanujan graphs. These graphs have many applications in the construction of explicit… (More)

Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the " function field " analogues, that is zeta functions of curves over finite fields and their generalizations. For… (More)