Effective Equidistribution in Diophantine Approximation : Theory, Interactions and Applications.

丢芬图近似是数论的一个分支，可以粗略地描述为对每个实数都可以被一个有理数任意近似的性质的定量分析。这个理论可以追溯到古希腊人和中国人，他们为了准确地预测行星和恒星的位置，使用了圆周率（3.14…）的近近值。今天，该理论与许多其他数学领域，如遍历理论、动力系统、概率论和分形几何，紧密地交织在一起。它还继续在应用于现实世界的问题中发挥重要作用，包括那些由计算机科学或电子通信，天线设计和信号处理等快速发展领域产生的问题。丢番图近似法与科学中其他学科之间的许多相互作用可以用用更规则的结构近似复杂结构的普遍需要来解释。因此，许多问题可以简化为对近似集分布的分析。在对这些集合所具有的规律性（即它们的等分布性质）有足够的了解之后，它们可以进一步得到解决。该研究项目旨在利用这一富有成效的观点来解决一些深刻而长期存在的问题，这些问题位于度量（即概率）数论、凸几何和丢芬图分析等主题的核心。其中一个目标与用有理数逼近相关量（例如，一个数及其立方）的问题有关。这导致流形上的丢番图近似领域，在这个领域中，大多数问题没有一般的理论可用。我们的目标是通过确定曲线上非常近似的点的分形维数，为一大类曲线发展这样的一般理论。这与计算靠近给定曲线的有界分母的有理点的数量有关，并且构成了一个非常活跃的研究领域。该项目还将涉及凸几何中的一个问题，可以松散地（但出奇地简单）描述如下：假设你站在森林中，所有的树干都有相同的（非常小的）尺寸。你能把它们放在这样一个位置上吗？它们彼此之间至少有一个单位距离，无论你站在哪里，无论你朝哪个方向看，你都永远看不到地平线。如果是这样，在以合适的方式排列树木时，可以保证的最小能见度是多少？潜在的深层问题是由Danzer（1965）提出的，并且与数学物理和数学准晶体理论中的其他问题密切相关。它仍然是开放的，将通过分析定义为树中心的点集的分布，用丢番图的方法来解决。最后，另一个目标将是回答一些与有效性问题相关的问题（即，已知结果成立，但不知道如何在任何具体示例中检查结果的问题）。更准确地说，重点将放在满足如下简单性质的数字序列上：无论何时，比如说，三个连续的数字是已知的，那么它们后面的数字可以从一个固定的简单规则中推导出来，比如将三个给定的数字相加并乘以常数。应该清楚的是，列表中前三个数字的数据和第四个数字的（固定）演绎规则决定了整个列表。然后可以证明，在有利条件下，该列表中出现的零的数量总是有限的。这个问题，在计算机科学的可决性的深层问题的核心，是能够找到一个范围，超过这个范围，可以保证所有的项都是非零的。这个问题是作为拟议项目的一部分来研究的。

Diophantine Approximation is a branch of Number Theory that can loosely be described as a quantitative analysis of the property that every real number can be approximated by a rational number arbitrarily closely. The theory dates back to the ancient Greeks and Chinese who used good approximations to the number pi (3.14...) in order to accurately predict the position of planets and stars. Today, the theory is deeply intertwined with many other areas of mathematics such as ergodic theory, dynamical systems, probability theory and fractal geometry. It also continues to play a significant role in applications to real world problems including those arising from computer science or from the rapidly developing areas of electronic communications, antenna design and signal processing.The many interactions between Diophantine Approximation and other disciplines in science can be explained by the universal need to approximate complex structures by more regular ones. Many problems can thus be reduced to the analysis of the distribution of sets of approximation. They can furthermore be solved upon having a good enough understanding of the regularity such sets enjoy (that is, their equidistribution properties). The research project aims at exploiting this fruitful point of view to tackle some deep and long-standing problems lying at heart of topics as varied as Metric (i.e. probabilistic) Number Theory, Convex Geometry and Diophantine Analysis.One of the goals is related to the problem of approximating dependent quantities (e.g., a number and its cube) by rationals. This leads to the domain of Diophantine Approximation on manifolds, where for most questions no general theory is available. Our aim is to develop such a general theory for a large class of curves by determining the fractal dimension of very well approximable points lying on them. This is related to the problem of counting the number of rational points with bounded denominators lying close to the given curves and constitutes an extremely active domain of research.The project will also be concerned with a question in Convex Geometry which can loosely (but surprisingly simply) be described as follows: suppose you stand in a forest where all tree trunks have the same (very small) size. Can you position them in such a way that they are at least a unit distance apart from each other and that no matter where you stand and what direction you look in, you will never be able to see the horizon? If so, what is the smallest visibility that can be guaranteed upon arranging the trees in a suitable way? The underlying deep problem is due to Danzer (1965) and is closely related to other questions in mathematical physics and in the theory of mathematical quasicrystals. It is still open, and will be addressed with Diophantine methods by analysing the distribution of the set of points defined as the centers of the trees.Finally, another goal will be to answer some questions related to problems of effectivity (that is, to problems where it is known that a result holds, but where it is not known how to check it in any concrete example). More precisely, the focus will be on sequences of numbers satisfying a simple property such as the following: whenever, say, three consecutive ones are known, then the number coming after them can be deduced from a fixed simple rule such as adding the three given numbers and multiplying them by constants. It should be clear that the data of the first three numbers in the list and of the (fixed) rule of deduction of the fourth one determines the entire list. It can then be shown that, under favorable conditions, the number of zeros appearing in this list is always finite. The question, at the heart of deep problems of decidability in Computer Science for instance, is to be able to find a range beyond which it can be guaranteed that all terms are nonzero. This question is studied as part of the proposed project.