Nilsystems play a key role in the structure theory of measure preserving systems, arising as
the natural objects that describe the behavior of multiple ergodic averages. This book is a …

Let $ p_n $ denote the $ n $ th prime. We prove that\[\max _ {p_ {n}\leqslant X}(p_ {n+ 1}-
p_n)\gg\frac {\log X\log\log X\log\log\log\log X}{\log\log\log X}\] for sufficiently large $ X …

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The Möbius disjointness conjecture of Sarnak states that the Möbius function does not
correlate with any bounded sequence of complex numbers arising from a topological …

Abstract Let g 0,…, gk: N→ D be 1-bounded multiplicative functions, and let h 0,…, hk∈ Z be
shifts. We consider correlation sequences f: N→ Z of the form f (a):= m→∞ 1 log ω m∑ xm/ω …

Let G (X) denote the size of the largest gap between consecutive primes below X. Answering
a question of Erdős, we show that G(X)\geqslantf(X)logXloglogXloglogloglogX(logloglogX) …

We prove that the Möbius function is linearly disjoint from an analytic skew product on the 2-
torus. These flows are distal and can be irregular in the sense that their ergodic averages …

We prove a structure theorem for multiplicative functions which states that an arbitrary
multiplicative function of modulus at most $1 $ can be decomposed into two terms, one that …

We establish the inverse conjecture for the Gowers norm over finite fields, which asserts
(roughly speaking) that if a bounded function f: V → C on a finite-dimensional vector space V …

A collection of integer sequences is jointly ergodic if for every ergodic measure preserving
system the multiple ergodic averages, with iterates given by this collection of sequences …