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Higher order Fourier analysis of multiplicative functions and applications
We prove a structure theorem for multiplicative functions which states that an arbitrary bounded multiplicative function can be decomposed into two terms, one that is approximately periodic andExpand
Multiple ergodic averages for three polynomials and applications
We find the smallest characteristic factor and a limit formula for the multi- ple ergodic averages associated to any family of three polynomials and polynomial fam- ilies of the formExpand
Ergodic averages of commuting transformations with distinct degree polynomial iterates
We prove mean convergence, as $N\to\infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f_1(T_1^{p_1(n)}x)\cdot\ldots\cdot f_\ell(T_\ell^{p_\ell(n)}x)$, where $p_1,\ldots,p_\ell$ areExpand
Multiple recurrence and convergence for sequences related to the prime numbers
For any measure preserving system (X, , μ,T) and A ∈ with μ(A) > 0, we show that there exist infinitely many primes p such that (the same holds with p − 1 replaced by p + 1). Furthermore, we show theExpand
Pointwise convergence for cubic and polynomial ergodic averages of non-commuting transformations
We study the limiting behavior of multiple ergodic averages involving several not necessarily commuting measure preserving transformations. We work on two types of averages, one that uses iteratesExpand
The polynomial multidimensional Szemerédi Theorem along shifted primes
AbstractIf $\vec q_1 ,...,\vec q_m $ : ℤ → ℤℓ are polynomials with zero constant terms and E ⊂ ℤℓ has positive upper Banach density, then we show that the set E ∩ (E − $\vec q_1 $ (p − 1)) ∩ … ∩ (EExpand
Equidistribution of sparse sequences on nilmanifolds
AbstractWe study equidistribution properties of nil-orbits (bnx)n∈ℕ when the parameter n is restricted to the range of some sparse sequence that is not necessarily polynomial. For example, we showExpand
A Hardy field extension of Szemerédi's theorem
Abstract In 1975 Szemeredi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common differenceExpand
Multiple recurrence and convergence for hardy sequences of polynomial growth
We study the limiting behavior of multiple ergodic averages involving sequences of integers that satisfy some regularity conditions and have polynomial growth. We show that for “typical” choices ofExpand
Multiple correlation sequences and nilsequences
We study the structure of multiple correlation sequences defined by measure preserving actions of commuting transformations. When the iterates of the transformations are integer polynomials we proveExpand