Problems on the interface of analysis, number theory and additive combinatorics

提出的研究旨在研究（密度）拉姆齐理论领域当前感兴趣的问题，并继续以前在离散谐波分析方面的工作。这些问题出现在不同的背景下，一个统一的主题是在绝对和相对设置的正密度集合中显示某些结构的存在。所提出的方法强调调和分析的经典技术、解析数论和加性组合的新方法之间的相互作用。

The proposed research aims to study problems of current interest in the area of (density) Ramsey theory and continue previous work in discrete harmonic analysis. The problems appear in various contexts, a unifying theme is to show the existence of certain structures in sets of positive density both in the absolute and relative settings. The proposed approaches emphasize the interplay between classical techniques of harmonic analysis, analytic number theory and new methods of additive combinatorics. A group of problems study linear and non-linear patterns, defined as solution sets of a family of diophantine equations, in the primes, prime-tuples and in more general sets satisfying certain pseudo-randomness conditions. Some of these problems are continuation of recent (joint) work on prime solutions of diophantine equations and on linear patterns in relative dense subsets of prime points. For polynomial equations we expect that our method extends the new situations such as the distribution of prime solutions when mapped into measure spaces. For the linear case, our method, based on a new sparse (or weighted) hyper-graph removal lemma to obtain multi-dimensional extension of the theorem of Green and Tao on arithmetic progression in the primes, may be utilized to address related problems both in the hypergraph and arithmetic settings (e.g. related arithmetic removal and regularity lemmas). Another family of problems, in the area of Euclidean Ramsey theory, concerns all large scale copies of a finite set of points in measurable subsets of positive density in Euclidean spaces. Here the difference is that one allows translations, dilations as well as rotations but one asks the existence of all "large" copies of a given pattern, measured in terms of some norm. Here we plan to continue on a very recent direction showing the possibility of obtaining such results even if the pattern have linear dependencies if the norm measuring the size of the pattern is chosen appropriately. In this case one needs to combine ideas from additive combinatorics and those of time-frequency analysis, as opposed to previous approaches based on classical Fourier analysis; we plan to pursue these connections further. Similar problems are proposed in the discrete settings, the underlying Euclidean space replaced by the integer lattice, where there is an additional number theoretic component related to the distribution of integer points on homogeneous spaces. The proposed project plans to continue previous work in discrete harmonic analysis on estimating maximal operators along polynomial sequences and related pointwise ergodic theorems for nilpotent group actions. Recently the proposer (in collaboration) have developed a new method for the closely related problem of estimating singular Radon transforms along polynomial sequences on step-2 groups. The approach has two new main ingredients; to use recursive estimates in place of almost-orthogonality arguments and to utilize techniques developed for systems of diophantine equations to understand the kernels of high powers of the singular operators. This latter problem is equivalent to showing that the many-fold product set of a polynomial sequence in a discrete nilpotent group is uniformly distributed, an extension of the so-called Waring-Tarry problem to the nilpotent setting, which may be of independent interest. We expect that our method extends to higher step groups, and can be further developed to estimate maximal operators and to study the pointwise distribution of polynomial sequences in measure spaces.