Recursion Theory and Diophantine Approximation

递归理论和丢番图近似

• 批准号: 1600441
• 负责人: Theodore Slaman
• 金额: $ 60万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600441&HistoricalAwards=false
• 关键词: Recursion; Theory; Diophantine; Approximation

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.

我们理解数学对象及其行为的一部分涉及了解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，pi 的十进制展开式中是否有无限多个 7 的问题询问了几何常数 pi 的属性（以 10 为基数表示）。这个数学基础研究项目研究了丢番图近似的有效且更普遍可定义的方面、可数组合数学中熟悉定理的元数学地位，以及相对可定义性的纯结构理论。人们可以将丢番图近似中的经典定理视为根据我们对实数的描述来断言实数的性质，例如波莱尔的几乎每个实数都是绝对正规的或罗斯的每个无理代数数都有无理指数2。博雷尔定理断言整数基展开的性质，而罗斯定理断言有理数逼近的性质。数论中的这一传统与递归理论中的可定义性研究之间存在着天然的亲和力。该研究项目将调查这些领域之间的联系。例如，实数的无理指数，或者更一般地说，它的马勒超越度量，与其二元展开式初始段的柯尔莫哥洛夫复杂度之间的确切关系是什么？其次，如果对于每个整数基 b，每个数字以渐近频率 1/b 出现，则实数是绝对正规的。还有哪些其他可能的渐近频率模式？最后一个问题还涉及 Furstenberg x2 x3 猜想形式的动力系统。