Máté Wierdl

Pointwise ergodic theorem along the prime numbers

The pointwise ergodic theorem is proved for prime powers for functions inLp,p>1. This extends a result of Bourgain where he proved a similar theorem forp>(1+√3)/2.

Oscillation and variation for the Hilbert transform

It is well known that this limit exists a.e. for all f ∈ L, 1 ≤ p < ∞. In this paper, we will consider the oscillation and variation of this family of operators as goes to zero, which gives extra information on their convergence as well as an estimate on the number of λ-jumps they can have. For earlier results on oscillation and variation operators in analysis and ergodic theory, including some historical remarks and applications, the reader may look in [2], [3], [5], [4], and [6]. For each fixed sequence (ti) ↘ 0, we define the oscillation operator ( H∗f ) (x) = ( ∞ ∑

Ergodic Theory and its Connections with Harmonic Analysis: Pointwise ergodic theorems via harmonic analysis

where 〈x〉 denotes the fractional part of x, that is 〈x〉 = x− [x], and |I| is the length of the interval I. In fact, Weyl went on to prove, in [Weyl, 1916], that the sequence α, 2α, 3α . . . is uniformly distributed mod 1. A bit less than twenty years later Vinogradov [cf. Ellison & Ellison, 1985] proved, as a byproduct of his solution of the ’odd’ Goldbach conjecture, that the sequence (pnα), where pn denotes the n-th prime number, is uniformly distributed mod 1. On the other hand, it is easy to see that for some irrational α the sequence (2α) is not uniformly distributed mod 1.

Oscillation and variation for singular integrals in higher dimensions

In this paper we continue our investigations of square function inequalities in harmonic analysis. Here we investigate oscillation and variation inequalities for singular integral operators in dimensions d > 1. Our estimates give quantitative information on the speed of convergence of truncations of a singular integral operator, including upcrossing and A jump inequalities.

Ergodic averaging sequences

We consider generalizations of the pointwise and mean ergodic theorems to ergodic theorems averaging along different subsequences of the integers or real numbers. The Birkhoff and Von Neumann ergodic theorems give conclusions about convergence of average measurements of systems when the measurements are made at integer times. We consider the case when the measurements are made at timesa(n) or ([a(n)]) where the functiona(x) is taken from a class of functions called a Hardy field, and we also assume that |a(x)| goes to infinity more slowly than some positive power ofx. A special, well-known Hardy field is Hardy’s class of logarithmico-exponential functions.The main theme of the paper is to point out that for a functiona(x) as described above, a complete characterization of the ergodic averaging behavior of the sequence ([a(n)]) is possible in terms of the distance ofa(x) from (certain) polynomials.

Oscillation in ergodic theory: Higher dimensional results

In this paper we continue our investigations of square function inequalities. The results in [9] are primarily one dimensional, and here we extend all the results to multi-dimensional averages. Our basic tool is still a comparison of the ergodic averages with various dyadic (reversed) martingales, but the Fourier transform arguments are replaced by more geometric almost orthogonality arguments.The results imply the pointwise ergodic theorem for the action of commuting measure preserving transformations, and give additional information such as control of the number of upcrossings of the ergodic averages. Related differentiation results are also discussed.

Chapter 12 – Combinatorial and Diophantine Applications of Ergodic Theory

This chapter discusses the combinatorial and diophantine applications of ergodic theory. Ergodic theory has its origin in statistical and celestial mechanics. In studying the long time behavior of dynamical systems, ergodic theory deals with such phenomena as recurrence and uniform distribution of orbits. On the other hand, Ramsey theory, a branch of combinatorics, is concerned with the phenomenon of preservation of highly organized structures under finite partitions, whereas diophantine analysis concerns itself with integer and rational solutions of systems of polynomial equations. It is also possible to derive from the polynomial Hales-Jewett theorem an analogue of the polynomial van der Waerden theorem that is valid in any commutative ring. Furthermore, central sets are an ideal object for Ramsey-theoretical applications. For example, central sets are not only large, that is, piecewise syndetic, but also are combinatorially rich and, particularly, contain IP sets and arbitrarily long arithmetic progressions.

The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters

In this paper we establish conditions on a sequence of operators which imply divergence. In fact, we give conditions which imply that we can find a set B of measure as close to zero as we like, but such that the operators applied to the characteristic function of this set have a lim sup equal to 1 and a lim inf equal to 0 a.e. (strong sweeping out). The results include the fact that ergodic averages along lacunary sequences, certain convolution powers, and the Riemann sums considered by Rudin are all strong sweeping out. One of the criteria for strong sweeping out involves a condition on the Fourier transform of the sequence of measures, which is often easily checked. The second criterion for strong sweeping out involves showing that a sequence of numbers satisfies a property similar to the conclusion of Kroneckers lemma on sequences linearly independent over the rationals.

Oscillation inequalities for rectangles

In this paper we extend previously obtained results on LP norm inequalities (1 < p < oo) for square functions, oscillation and variation operators, with Z actions, to the case of Zd actions. The technique involves the use of a result about vector valued maximal functions, due to Fefferman and Stein, to reduce the problem to a situation where we can apply our previous results.

Counting in Ergodic Theory

A number of phenomenon in ergodic theory can be understood by counting occurrences of various events. In Section 2, counting of levels shows how large deviation theorems immediately imply control of series like square functions. This leads to some unusual rate results for averages somewhat like the ones in the usual ergodic theorem. In Section 3, counting of jumps for martingales leads to a technique to show that the jump inequalities previously obtained for ergodic averages are the best possible ones. See also Ivanov [9] and Kachurovskii [13] for closely related results. In Section 4, counting the leading edge of the ergodic average gives an ergodic theorem due to Assani [2, 3] in Lp for p > 1 together with some improvements as a result of the theorems in Section 2. The theme of counting occurrences of various sorts for ergodic averages is seen in all of the sections.