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

拟议的研究旨在研究（密度）拉姆齐理论领域的当前兴趣的问题，并继续在离散谐波分析以前的工作。这些问题出现在不同的背景下，一个统一的主题是显示在绝对和相对设置中的正密度集合中存在某些结构。所提出的方法强调调和分析，解析数论和添加剂组合学的新方法的经典技术之间的相互作用。 一组问题研究线性和非线性模式，定义为一个家庭的丢番图方程，在素数，素数元组和更一般的集合满足一定的伪随机性条件的解决方案集。其中一些问题是延续最近（联合）的工作，主要解决方案的丢番图方程和线性模式的相对密集子集的素数点。对于多项式方程，我们期望我们的方法扩展了新的情况，如映射到测度空间时素解的分布。 对于线性情形，我们的方法基于一个新的稀疏（或加权）超图移除引理，得到了绿色和陶关于素数算术级数定理的多维推广，可以用来解决超图和算术设置中的相关问题（例如相关的算术移除引理和正则引理）. 另一个家庭的问题，在该地区的欧几里得拉姆齐理论，涉及所有大规模副本的有限集点的可测子集的正密度在欧几里得空间。这里的区别在于，人们允许平移、膨胀和旋转，但人们要求存在给定模式的所有“大”副本，以某种规范来衡量。在这里，我们计划继续在最近的方向上显示获得这样的结果的可能性，即使模式具有线性依赖性，如果适当选择测量模式大小的范数。在这种情况下，需要结合联合收割机的想法，从添加剂组合和时频分析，而不是以前的方法基于经典的傅立叶分析，我们计划进一步追求这些连接。类似的问题提出了在离散设置，基本的欧几里得空间取代的整数格，其中有一个额外的数论组件有关的分布的整数点齐次空间。 拟议的项目计划继续以前的工作，在离散调和分析估计最大运营商沿着多项式序列和相关的逐点遍历定理的幂零群行动。最近的提议者（在合作）已经开发了一种新的方法，估计奇异Radon变换沿着多项式序列的步骤-2组密切相关的问题。 该方法有两个新的主要成分;使用递归估计代替几乎正交参数，并利用丢番图方程系统开发的技术来了解奇异算子的高功率的内核。后一个问题等价于证明一个离散幂零群中的多项式序列的多重乘积集是均匀分布的，这是所谓的Waring-Tarry问题在幂零集上的扩展，这可能是独立的。我们希望我们的方法可以推广到更高阶的群，并可以进一步发展到估计极大算子和研究测度空间中多项式序列的点态分布。

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.