Recursion Theory and Diophantine Approximation

我们理解数学对象及其行为的一部分涉及到我们指定这些对象的方式以及我们用来研究它们的方法。例如，在π的十进制展开式中是否有无穷多个7的问题询问了几何常数π的一个属性，该属性以我们的十进制表示形式表示。数学基础中的这个研究项目研究了丢番图近似的有效性和更普遍的可定义性，可数组合学中熟悉定理的元数学状态，和相对可定义性的纯结构理论。人们可以把丢番图逼近中的经典定理，如Borel的几乎每个真实的数都是绝对正规的或Roth的每个无理代数数都有无理指数2，看作是根据我们对它们的描述来断言真实的数的性质。波莱尔定理断言了一个由整数基展开的性质，而罗斯定理断言了一个由有理数逼近的性质。数论中的这一传统与递归论中的可定义性研究之间有着天然的密切关系。这个研究项目将调查这些领域之间的联系。例如，一个真实的数的无理指数，或者更一般地说，它的马勒超越测度，与它的二进制展开式的初始段的柯尔莫哥洛夫复杂度之间的确切关系是什么？第二，一个真实的数是绝对正规的，如果对于每一个整数基B，每个数字出现的渐近频率为1/B。渐近频率的其他模式是可能的吗？这最后一个问题也涉及到动力系统的形式弗斯滕伯格的x2 x3猜想。

Part of our understanding mathematical objects and their behaviors involves knowing the ways by which we specify those objects and the means that we employ to study them. For example, the question of whether there are infinitely many 7's in the decimal expansion of pi asks about a property of the geometric constant pi as expressed in our base-10 representation of it. This research project in the foundations of mathematics investigates the effective, and more generally definable, aspects of Diophantine approximation, the meta-mathematical status of familiar theorems in countable combinatorics, and the pure structure theory of relative definability. One can view classical theorems in Diophantine approximation, such as Borel's almost-every real number is absolutely normal or Roth's every irrational algebraic number has irrationality exponent 2, as asserting properties of real numbers in terms of our descriptions of them. Borel's theorem asserts a property of expansions by integer bases, and Roth's theorem asserts a property of approximation by rational numbers. There is a natural affinity between this tradition in number theory and the study of definability in recursion theory. This research project will investigate the connections between these areas. For example, what is the exact relationship between the irrationality exponent of a real number, or more generally its Mahler transcendence measures, and the Kolmogorov complexity of the initial segments of its binary expansion? Second, a real number is absolutely normal if for every integer base b, each digit appears with asymptotic frequency 1/b. What other patterns of asymptotic frequencies are possible? This last question also touches dynamical systems in the form of Furstenberg's x2 x3 conjecture.