RESEARCH STATEMENT WENBO SUN

RESEARCH STATEMENT WENBO SUN My research focus is on ergodic theory, especially on problems that are at the intersection of dynamics, combinatorics, harmonic analysis, and number theory. To be more precise, I work on understanding the structure theorems for dynamical systems, and using them to (i) address problems within pure ergodic theory, (ii) solve problems in ergodic theory but motivated by outside areas, and (iii) export results and techniques from ergodic theory to apply them to outside areas. In most of my works, while the main results and techniques are in ergodic theory, much of the motivation and interest comes from allied areas. 1. background Let (X, X, µ, T 1,..., T d ) be a measure preserving system (meaning that (X, X, µ) is a probability space and each T i is a transformation acting on X such that µ(ti 1 A) = µ(a) for all A X). 1 A multiple average refers to an expression similar to the following form: 1 N 1 (1.1) f 1 (T n 1 N x) f 2(T n 2 x) f d(t n d x), i=0 where f 1,..., f d are bounded functions on X. The study of (1.1) can be traced back to Furstenberg [Fur], who proved that (1.2) lim inf N 1 N 1 N i=0 µ(a T n A T dn A) > 0 for all A X with µ(a) > 0. By the Furstenberg Correspondence Principle [Fur], this result implies Szemerédi s Theorem: every subset E N with positive upper Banach density (meaning that d I E (E) = lim sup I > 0) contains arbitrarily long arithmetic I progressions. In [HK2], Host and Kra proved that if T i = T i, then the L 2 limit of (1.1) exists as N. In this paper, a structure theorem was introduced. Roughly speaking, for every k N, they defined a k-th structured property and a k-th uniformity property such that for every measure preserving system (X, X, µ, T) and every f L (µ), we can decompose f as (1.3) f = f s + f u, 1 Usually we assume that the transformations are commuting with each other, i.e. T i T j = T j T i. 1

2 WENBO SUN where f s has the k-th structured property and f u has the k-th uniformity property. They then used this structure theorem to prove the L 2 convergence of (1.1). The structure theorem proved to be a powerful tool in the study of problems related to multiple averages. Nowadays, the structure theorem has been generalized to various settings, and many applications of them are revealed in combinatorics and number theory. Following the work of Host and Kra [HK2], fruitful results on the L 2 convergence of (1.1) has been obtained in recent years (see [Aus, H, T, W, Z] for example). Most of my work are initialized in the study of multiple average problems and structure theorems, and then radiate to questions from various areas. In the following sections, I will explain the role of structure theorems to addressing problems within pure ergodic theory, in ergodic theory but motivated by outside areas, and in exporting results and techniques from ergodic theory to apply them to outside areas. 2. Addressing problems within pure ergodic theory 2.1. Pointwise convergence for distal systems. While the L 2 convergence of (1.1) has been widely studied as is mentioned in the introduction, little is known about the pointwise convergence, i.e. whether (1.1) converges for a.e. x X as N. Bourgain [Bou] gave an affirmative answer to this question for the case d = 2, T 1 = T a, T 2 = T b, a, b Z using methods from harmonic analysis. Recently, by a new method using topological models, Huang, Shao and Ye [HSY1] proved the pointwise convergence of (1.1) for T i = T i, i = 1,..., d under the assumption that (X, µ, T) is distal (see [HSY1] for the definition). In a series of joint work with S. Donoso [4, 7, 6], we imporved the result of [HSY1] to systems with commuting transformations (among other pointwise convergence results): Theorem 2.1 (S. Donoso and S. [6]). Let (X, X, µ, T 1,..., T d ) be a distal system with commuting transformations. Then for all f 1,..., f d L (µ), the limit of (1.1) as N exists for µ-a.e. x X. The method to prove Theorems 2.1 is to (a) construct an appropriate topological model for the system X; (b) and then use this model to study the corresponding averages. This method generalizes the ideas of Huang, Shao and Ye in [HSY1, HSY2], where the same questions were studied for the case T i = T i. The techniques used in step (b) were generalized from the methods used in [4, 7, HSY1, HSY2]. However, it is in step (a) where major innovations are involved. While the methods in [HSY1, HSY2] rely heavily on the structure theorem of Host and Kra [HK2] for the case T i = T i, there is no explicit structure theorem in the general setting. To overcome this difficulty, we use the theories developed by Austin [Aus] and Host [H] to replace the structure theorem, and then use them to construct suitable topological models to prove Theorems 2.1. Based on this work, my next goal is to study Theorem 2.1 for the non-distal case. In the mean time, I will seek applications of Theorem 2.1 to other questions (some of which are already in progress).

RESEARCH STATEMENT 3 2.2. Other results. Based on the existing knowledge, I also studied other problems in pure ergodic theory in the past, which includes the multiple recurrence and L 2 convergence for generalized linear polynomials along shifted primes [11], multiple pointwise convergence for sublinear functions [2], and under-/over-recurrence properties for various types of mixing systems [1]. In the future, I will continue to expand my knowledge in pure ergodic theory, and to study broader questions in this area. 3. Addressing problems in ergodic theory motivated by outside areas 3.1. Partition regularity for quadratic equations. In [EG] Erdös and Graham asked the following question: for any finite partition of integers Z = m j=1 A j, are there non-zero x, y, z A j with x 2 y 2 = z 2 for some j? For the case where x 2 y 2 = z 2 is replaced by ax + by = z 2, many results are known (see for example [BL, Fur, KS, Sar]). Recently, Frantzikinakis and Host [FH] showed that there exist x, y A j, n Z with x y such that 16x 2 + 9y 2 = n 2 for some j and n Z. In fact, their result covered a more general class of quadratic equations. But we remark that the equation x 2 y 2 = n 2 was not covered. In [10], we showed that the result of [FH] holds for a more general class of quadratic equations (including x 2 y 2 = n 2 ) if one replace the integer domain Z with the Gaussian integers Z[i]. To avoid technique issues, we only give a particular case of the full result: Theorem 3.1 (S. [10]). For any finite partition Z[i] = m j=1 A j of Z[i], there exist 1 j m and non-zero x, y A j with x y such that for some n Z[i]. x 2 y 2 = n 2 A function χ: Z[i] C is multiplicative if χ(αβ) = χ(α)χ(β) for all α, β Z[i], (α, β) = 1. The idea of proving Theorem 3.1 is to use a variation of the Furstenburg Correspondence Principle [Fur] to convert Theorem 3.1 to the study of a variation of (1.1), with the key ingredient being the following structure theorem for multiplicative functions: Theorem 3.2 (S. [10]). Let χ: Z[i] C be a multiplicative function. Then for all k 3, χ can be written as χ = χ u + χ s + χ e, where the χ s is almost periodic, χ u has the k-th uniformity property, and χ e is an error term. 2 A similar result for multiplicative functions on Z was proved in [FH], and we refer the precise statements of these structure theorems to [FH, 10]. This theorem is the main innovation of [10], and it require the development of many new tools to prove it. For the future plan of this problem, I will study the generalization of Theorem 3.1 for other quadratic equations over a different domain Z[ d]. I will also attempt to generalize Theorem 3.2 for k > 3, which is of independent interest and has other potential applications. 2 Note that we only proved this theorem for the case k 3 and the general case is still open. Even so, this structure theorem has an interesting combinatorial application (Theorem 3.1).

4 WENBO SUN 3.2. Optimal lower bound problem. The Poincaré recurrence theorem states that for every measure preserving system (X, X, µ, T) and every A X with µ(a) > 0, the set {n Z: µ(a T n A) > 0} is infinite. A quantitative version of it was provided by Khintchine [K], who showed that for every ɛ > 0, the set {n N: µ(a T n A) > µ(a) 2 ɛ} is syndetic, meaning that it has bounded gaps. Let (X, X, µ, T 1,..., T d ) be a measure preserving system and F : [0, 1] [0, ) be a function. We say that F is good for (T 1,..., T d ) if the set {n N: µ(a T n 1 A T n d A) > F(µ(A)) ɛ} is syndetic for all ɛ > 0. It is natural to ask if there is a quantitative version of Furstenberg s recurrence theorem (1.2): Question 3.3. Let F(u) = u d+1. Is F good for (T 1,..., T d ) for all ergodic measure preserving systems (X, µ, T 1,..., T d )? It was shown in [BHK] that if (X, µ, T) is an ergodic system, then F(u) = u d+1 is good for (T, T 2,..., T d ) if n 3, and is not good if n 4. In general, F(u) = u d+1 is good for (T a 1, T a 2,..., T a d ) for n 2, and for n = 3, a 1 + a 2 = a 3 [Fra1]. In [3], we studied the case n = 3 and a 1 + a 2 a 3. Using the structure theorem, we showed that this question is almost equivalent to a problem in combinatorics. For concision I will only explain a special case. Let (a 1, a 2, a 3 ) = 2, 3, 4 and L: Z 4 Z, L(x, y, z, w) = x + 8y 3z 6w. For E {1,..., N}, let D N (L, E) = {(x, y, z, w) E 4 : L(x, y, z, w) = 0} /N 3 denote the density of the solutions to L = 0 in E, and d N (E) = E /N denote the density of E in {1,..., N}. Among other results, we have Theorem 3.4 (S. Donoso, A. Le, J. Moreira and S. [3]). Let l > 4 and L(x, y, z, w) = x + 8y 3z 6w. Suppose that for every large enough N and every subset E {1,..., N}, D N (L, E) d N (E) l. Then for every ergodic system (X, X, µ, T), F(u) = u l is good for (T 2, T 3, T 4 ). It is worth noting that partial results in the opposite direction was also obtained in [Fra1]. In [3], we showed that this condition is almost necessary and sufficient, and a similar result holds for all a 1, a 2, a 3 (see [3] for details). The ideas of the proof is to show that lim sup N M 1 S [M, N) n S [M,N) µ(a T 2n T 3n T 4n A) µ(a) l for some Bohr 0 set S N. By the structure theorem, we may assume that X is a nilsystem. Then using the algebraic structure of X, we manage to connect this expression with the combinatorial quantity D N (L, E). It is also natural to ask Question 3.3 for systems with commuting transformations. It was proved by Chu [C] that F(u) = u 4 is good for (T 1, T 2 ). In the joint work with S. Donoso [8], among other results, we showed that F(u) = u 4 is optimal:

RESEARCH STATEMENT 5 Theorem 3.5 (S. Donoso and S. [8]). Let F : [0, 1] [0, ), F(u) = u l. Then for any 0 < l < 4, there exists a system with commuting transformations (X, X, µ, T 1, T 2 ) such that (T 1, T 2 ) is not good for F. This result combined with [C] provide an almost complete answer to the goodness for systems with more than one transformations. The proof of Theorem 3.5 relies on the discovery of its connection with a problem in combinatorics, which is the main innovation of the paper. We say that a set E {1,..., N} 3 is three-point-free if (x, y, z ), (x, y, z), (x, y, z) E implies that x = x, y = y, z = z. By clever construction of the systems, one can reduce the proof of Theorem 3.5 to the construction of a large set E which is three-point-free. We then use tools in combinatorics to construct such sets, and thus to complete the proof. Up to now, Question 3.3 is only answered for some special cases, and there are still many open questions related to it. These questions will be our next goal. By studying such questions, one can reveal more connections between ergodic theory and combinatorics. 4. Exporting results from ergodic theory to outside areas 4.1. Equidistribution for dilated curves. The techniques from the structure theorem can be applied to study equidistribution problems on nilmanifolds. Let X = G/Γ, where G is a (k-step) nilpotent group and Γ is a discrete and coocmpact subgroup of G. Let X and µ be the Borel σ-algebra and the Haar measure of X. We say that the probability space (X, X, µ) is a (k-step) nilmanifold. A family of probability measures (µ t ) t 0 on X is equidistributed on X if lim t µ t = µ in the weak topology. Given a measurable map φ: (0, 1) g (g is the Lie algebra of G), x 0 X and t 0, let µ φ,x0,t (called the dilation of φ) be the measure on X defined by 1 f dµ φ,x0,t = f (exp(tφ(u)) x 0 ) du X 0 for all f C(X). It was proved by Björklund and Fish [BF] that if X is a Heisenberg nilmanifold and φ is a polynomial map, then (µ φ,x0,t) t 0 is equidistributed on X. They conjectured that a similar result holds for any nilmanifold. In the work joint with Kra and Shah [9], we answered this conjecture and proved an even stronger result. Let λ denote the Lebesgue measure on (0, 1) throughout this section. For any horizontal character χ of X (see [9] for definition), let dχ: g R denote its differential. We have Theorem 4.1 (B. Kra, N. Shah and S. [9]). Let (X, X, µ) be a nilmanifold and φ: (0, 1) g be a map such that φ (u) exists for λ-a.e. u (0, 1). Suppose that for every horizontal character χ of X, (4.1) φ (u) ker(dχ) for λ-a.e. u (0, 1), then (µ φ,x0,t) t 0 is equidistributed on X for all x 0 X. Moreover, if φ is analytic, then (4.1) is also a necessary condition.

6 WENBO SUN We also obtained results more general than Theorems 4.1, where µ φ,x0,t is replaced with µ φ,x0,ρ t for some polynomial map ρ t of t. Moreover, motivated by the work of Chaika and Hubert [CH], we also studied the weak equidistribution and obtained necessary and sufficient conditions for dilation of curves on nilmanifolds (see [9]), as well as for dilation of R m -translations on distal systems (see [12]). To prove Theorem 4.1, we borrowed the tools from ergodic theory. We first use existing tools in ergodic theory to analyze the equidstribution properties of linear curves. Then we approximate an arbitrary curve φ with a piecewise linear curve ψ to get the conclusion we want. The main difficulty is that for each fixed approximation ψ of φ, the difference between µ φ,x0,t and µ ψ,x0,t explode when t. In order to control this difference, all the tool we developed has to be rather quantitative. In the next few years, I will try to export the tools in ergodic theory to even broader areas. One of the good questions to begin with is to study analogs of Theorems 4.1 for general measure preserving systems. This question will require more tools related to structure theorem, and will also bring these tools to areas outside ergodic theory. 4.2. Properties for tiling systems. The techniques in ergodic theory can also be exported to the study of topological tiling systems. As an example, in the joint work with S. Donoso [5], we give a structural description of the Morse tiling system, and computed the automorpism group of the minimal Robinson tiling system. Proposition 4.2 (S. Donoso and S. [5]). The Morse tiling system is a factor of a product system. Proposition 4.3 (S. Donoso and S. [5]). The automorpism group of the minimal Robinson tiling system is spanned by vertical and horizontal shifts. A topological dynamical system (X, T 1,..., T d ) consists of a compact topolocigal space X and continuous transformations T i : X X acting on X. Analogous to the measure theoretic structure theorem, Host, Kra and Maass [HKM] introduced an equivalent relation RP [d] (X) for topological systems (X, T) with a single transformation. They showed that the factor X/RP [d] (X) is the structured part of a topological system. In [5], we extend the relation RP [2] (X) to a new one RP T1,T 2 (X) for systems (X, T 1, T 2 ) with commuting transformations, and provided an explicit description of the structured part of such systems: Theorem 4.4 (S. Donoso and S. [5]). Let (X, T 1, T 2 ) be a system with two commuting transformations. Then (X/RP T1,T 2 (X), T 1, T 2 ) is a factor of a system of the form (Y Z, S id, id T). Moreover, X/RP T1,T 2 (X) is the largest factor of X having this property. Propositions 4.2 and 4.3 are applications of Theorem 4.4. In fact, since the relation RP T1,T 2 (X) is trivial for the Morse tiling system, Theorem 4.4 implies Proposition 4.2. Note that the relation RP T1,T 2 (X) is preserved under automorpisms, another direct computation yields Proposition 4.3. It is natural to ask generalizations of Theorem 4.4 for systems with more than two transformations, and this will be our next goal to study.

RESEARCH STATEMENT 7 References [Ada] T. Adams. Over recurrence for mixing transformations. arxiv: 1701.04345. [Aus] T. Austin. On the norm convergence of nonconventional ergodic averages. Ergodic Theory and Dynamical Systems, 30 (2010), no. 2, 321-338. [Ber] V. Bergelson. Ergodic Ramsey Theory - an update. In Ergodic Theory of Z d -actions (edited by M. Pollicott and K. Schmidt). London Math. Soc. Lecture Note Series 288 (1996), 1-61. [BHK] V. Bergelson, B. Host and B. Kra. Multiple recurrence and nilsequences. With an appendix by Imre Ruzsa, Invent. Math. 160 (2005), no. 2, 261-303. [BHMP] V. Bergelson, B. Host, R. McCutcheon and F. Parreau. Aspects of uniformity in recurrence. Colloq. Math. 84/85 (2000), no. 2, 549-576. [BL] V. Bergelson and A. Leibman. Polynomial extensions of van der Waerdens and Szemerédis theorems. J. Amer. Math. Soc. 9 (1996), no. 3, 725-753. [BLS] V. Bergelson, A. Leibman and Y. Son. Joint ergodicity along generalized linear functions. Ergodic [BF] Theory and Dynamical Systems, 36 (2016), no. 7, 2044-2075. M. Björklund and A. Fish. Equidistribution of dilations of polynomial curves in nilmanifolds. Proc. Amer. Math. Soc. 137 (2009), no. 6, 2111-2123. [BFW] M. Boshernitzan, N. Frantzikinakis and M. Wierdl, Under recurrence in the Khintchine recurrence theorem. Isr. J. Math. 222 (2017), 815-840. [Bou] J. Bourgain. Double recurrence and almost sure convergence. J. Reine Angew. Math. 404 (1990), 140-161. [CH] [C] [EG] J. Chaika and P. Hubert. Circle averages and disjointness in typical flat surfaces on every Teichmüller disc. Bull. Lond. Math. Soc., 2017. Q. Chu. Multiple recurrence for two commuting transformations. Ergodic Theory and Dynamical Systems, 31 (2011), no. 3, 771-792. P. Erdös and R. Graham. Old and new problems and results in combinatorial number theory. Monographies de L Enseignement Mathematique, 28. Universite de Geneve, L Enseignement Mathematique, Geneva, 1980. [Fra1] N. Frantzikinakis. Multiple ergodic averages for three polynomials and applications. Transactions of the American Mathematical Society, 360 (2006) no. 10: 5435-5475. [Fra2] N. Frantzikinakis. Multiple recurrence and convergence for Hardy field sequences of polynomial growth. J. Anal. Math. 112 (2010), 79-135. [FH] N. Frantzikinakis and B. Host. Higher order Fourier analysis of multiplicative functions and applications. Journal of the American Mathematical Society, (2014), 30. [FHK] N. Frantzikinakis, B. Host and B. Kra. The polynomial multidimensional Szemerédi the eorem along [Fur] shifted primes. Isr. J. Math., 194 (2011), no. 1: 331-348. H. Furstenberg. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Analyse Math. 71 (1977), 204-256. [GT] B. Green and T. Tao. Linear equations in primes. Ann. of Math. (2) 171 (2010), no. 3, 1753-1850. [GTZ] B. Green, T. Tao and T. Ziegler. An inverse theorem for the Gowers U s+1 -norm. Ann. of Math. (2) 176 (2012), no. 2, 1231-1372. [H] B. Host. Ergodic seminorms for commuting transformations and applications. Studia Math. 195 (2009), no. 1, 31-49. [HK1] B. Host and B. Kra. Convergence of polynomial ergodic averages. Isr. J. Math. 149 (2005), 1-19. [HK2] B. Host and B. Kra. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161 (2005), no. 1, 397-488. [HKM] B. Host, B. Kra and A. Maass. Nilsequences and a structure theory for topological dynamical systems. Adv. Math. 224 (2010), no. 1, 103-129. [HSY1] W. Huang, S. Shao and X. Ye. Pointwise convergence of multiple ergodic averages and strictly ergodic models. to appear J. Analyse Math., arxiv:1406.5930.

8 WENBO SUN [HSY2] W. Huang, S. Shao and X. Ye. Strictly ergodic models and the convergence of non-conventional pointwise ergodic averages. Mathematics, (2013). [KS] A. Khalfalah and E. Szemerédi. On the number of monochromatic solutions of x + y = z 2. Combin. Probab. Comput. 15 (2006), no. 1-2, 213-227. [K] A. Khintchine. Eine Verschärfung des Poincaréschen Wiederkehrsatzes. (German), Compositio Math. 1 (1935), 177-179. [Kou] A. Koutsogiannis. Closest integer polynomial multiple recurrence along shifted primes. Ergodic Theory and Dynamical Systems, 38 (2018), no. 2, 666-685. [L] A. Leibman. Convergence of multiple ergodic averages along polynomials of several variables. Isr. J. Math. 146 (2005), 303-316. [Sar] A. Sárközy. On difference sets of integers. III. Acta Math. Acad. Sci. Hungar. 31 (1978), no. 3-4, 355-386. [SY] S. Shao and X. Ye. Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence. Adv. Math. 231 (2012), no. 3-4, 1786-1817. [T] T. Tao. Norm convergence of multiple ergodic averages for commuting transformations. Ergodic Theory and Dynamical Systems 28 (2008), no. 2, 657-688. [W] M. Walsh. Norm convergence of nilpotent ergodic averages. Ann. of Math. (2) 175 (2012), no. 3, 1667-1688. [Z] T. Ziegler. A non-conventional ergodic theorem for a nilsystem. Ergodic Theory Dynam. Systems 25 (2005), no. 4, 1357-1370. [1] T. Adams, V. Bergelson and W. Sun. Under- and over-independence in measure preserving systems. submitted. arxiv: 1807.02966. [2] S. Donoso, A. Koutsogiannis and W. Sun. Pointwise multiple averages for sublinear functions. Ergodic Theory and Dynamical Systems, to appear. arxiv: 1711.01513. [3] S. Donoso, A. Le, J. Moreira and W. Sun. Optimal lower bounds for multiple recurrence. submitted. arxiv: 1809.06912. [4] S. Donoso and W. Sun. A pointwise cubic average for two commuting transformations. Israel Journal of Mathematics, 216 (2016), no. 2, 657-678. [5] S. Donoso and W. Sun. Dynamical cubes and a criteria for systems having product extensions. Journal of Modern Dynamics, 9 (2015), no. 1, 365-405. [6] S. Donoso and W. Sun. Pointwise convergence of some multiple ergodic averages. Advances in Mathematics, 330 (2018), 946-996. [7] S. Donoso and W. Sun. Pointwise multiple averages for systems with two commuting transformations. Ergodic Theory and Dynamical Systems, (2017) 1-26. doi:10.1017/etds.2016.127. [8] S. Donoso and W. Sun. Quantitative multiple recurrence for two and three transformations. Israel Journal of Mathematics, (2018), 1-15. [9] B. Kra, N. Shah and W. Sun. Equidistribution of dilated curves on nilmanifolds. Journal of the London Mathematical Society, doi:10.1112/jlms.12156. [10] W. Sun. A structure theorem for multiplicative functions over the Gaussian integers and applications. Journal d Analyse Mathematique, 134 (2018), no. 1, 55-105. [11] W. Sun. Multiple recurrence and convergence for certain averages along shifted primes. Ergodic Theory and Dynamical Systems. 35 (2015), no. 5, 1592-1609. [12] W. Sun. Weak ergodic averages over dilated measures. submitted. arxiv: 1809.06916. Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus OH, 43210-1174, USA E-mail address: sun.1991@osu.edu