Ergodic theory and additive combinatorics

A beautiful theorem of Szemeredi states that there are arbitrarily long arithmetic progressions in subsets of the integers of positive density. The ergodic theoretic approach to this problem was pioneered by Furstenberg in his ergodic theoretic proof of Szemeredi's theorem. Furstenberg observed that the question of finding patterns in subsets of the integers is intimately related to the phenomenon of multiple recurrence in ergodic theory. He then proved a structure theorem related to the asymptotic behavior of multiple ergodic averages associated with arithmetic progressions. A finer structure theorem was proved much later. Quite remarkably, it revealed a connection between the question of arithmetic progressions in large subsets and dynamical systems of a special algebraic nature called nilsystems - translations on homogeneous spaces of nilpotent groups.

A major breakthrough in the combinatorial approach to the question of arithmetic progression came with the work of Gowers, introducing uniformity norms which measure a certain kind of psuedorandomness: a Uk uniform set in a finite abelian group contains roughly the expected number of linear configurations as one would expect in a random subset of the same density.
Gowers then proved a local structure theorem for the uniformity norms, leading to new quantitative bounds for Szemeredi's theorem.

The role of nilsystems in the ergodic theoretic approach to the question of arithmetic progressions inspired Green and Tao to conjecture that a similar role should be played by nilsystems in the global behavior of the Gowers norms. This question was first studied in the context of finite field geometry, and resolved in a series of papers by the author in joint work with Bergelson and Tao, and with Tao.
In subsequent joint work with Green and Tao the conjecture was completely resolved in the cyclic group setting. This result has far reaching application in number theory. Combined with earlier work of Green and Tao it led to the proof of a well-known conjecture of Hardy and Littlewood regarding solutions of systems of affine linear equations of finite complexity in the prime numbers. Further applications have since been found in many other areas including arithmetic geometry, algebraic geometry

The aim of the proposal was to build on our previous work to extend the frontier of current knowledge.

I have recruited a number of excellent graduate students and postdocs working o problems related the proposal. We have made substantial progress on several of the key problems in the proposal and have discovered surprising new connections with algebraic geometry.

We describe some highlights of developments by the research group funded by the grant:

Together with T. Tao we established a number of "concatenation theorems" that assert, roughly speaking, that if a function exhibits "Gowers anti-uniform",behaviour in two different directions separately, then it also exhibits the same behavior (but at higher degree) in both directions jointly. Among other things, this allows one to control averaged local Gowers uniformity norms by global Gowers uniformity norms. In a sequel to this paper, we will apply such control to obtain asymptotics for "polynomial progressions" n+P1(r),…,n+Pk(r) in various sets of integers, such as the prime numbers.

Together with D. Kazhdan we discover new interesting connections between algebraic geometry and additive combinatorics. We define a new algebraic notion of Approximate cohomology and calculate it in a special case using techniques from additive combinatorics. We prove several important properties of high rank varieties; in particular we give a new proof Stilman’s conjecture in algebra using techniques from finite fields.

Together with my graduate student A. Lampert we developed the notion of relative rank for polynomials and proved important results relating relative rank and bias. In particular we proved the any variety defined over a field which is either finite or of bounded is the locus of r polynomials of bounded degree contains a high (relative) rank sub variety of codimension that is bounded polynomial in terms of the number of polynomials genre. We expect this notion to have strong significance in diophantine geometry.

My graduate student O. Shalom developed structure theory for the Host-Kra uniformity norms for the action of groups that are infinitely generated and but with unbounded torsion, extending results of Bergelson-Tao-Ziegler.

In joint work with K. Matomäki, M. Radziwiłł, T. Tao, and J. Teräväinen, we prove uniformity results for bounded multiplicative functions in short intervals on average. As a consequence we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of Liouville over short polynomial progressions, which in the case of linear polynomials yields a new averaged version of Chowla's conjecture.

Periodic Reporting for period 4 - ErgComNum (Ergodic theory and additive combinatorics)

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