Fekete引理中的有效收敛性及信息论中相关组合问题
On Effective Convergence in Fekete's Lemma and Related Combinatorial Problems in Information Theory
AI总结：
本文从实数算术层级角度分析Fekete引理，证明其极限值一般不可计算，并刻画了非负序列下极限值可有效计算的条件及其收敛模，进而讨论了对信息论问题的影响。
摘要：
Fekete引理是组合数学中一个众所周知的结果，它表明与实数的超加性和次加性序列相关的极限值的存在性。本文从Zheng和Weihrauch提出的实数算术层级的角度分析Fekete引理，并将结果置于信息论的语境中。我们引入与超加性和次加性序列相关的特殊集合，并证明它们与\(Σ_1\)和\(Π_1\)有效等价。利用Zheng和Weihrauch所建立理论中的方法，我们随后证明由Fekete引理产生的极限值通常不是可计算数。对于额外满足非负性的序列，我们刻画了在何种条件下相应的极限值可以被有效计算，并研究了相应的收敛模。作为辅助结果，我们证明了一个关于可计算数序列与有理数序列之间结构差异的定理。最后，我们讨论了我们的发现如何影响信息论中的常见问题。
英文摘要：
Fekete's lemma is a well known result from combinatorial mathematics that shows the existence of a limit value related to super- and subadditive sequences of real numbers. In this paper, we analyze Fekete's lemma in view of the arithmetical hierarchy of real numbers by Zheng and Weihrauch and fit the results into an information-theoretic context. We introduce special sets associated to super- and subadditive sequences and prove their effective equivalence to \(Σ_1\) and \(Π_1\). Using methods from the theory established by Zheng and Weihrauch, we then show that the limit value emerging from Fekete's lemma is, in general, not a computable number. Given a sequence that additionally satisfies non-negativity, we characterize under which conditions the associated limit value can be computed effectively and investigate the corresponding modulus of convergence. Subsidiarily, we prove a theorem concerning the structural differences between computable sequences of computable numbers and computable sequences of rational numbers. We close the paper by a discussion on how our findings affect common problems from information theory.
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