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Unprovability of sharp versions of Friedman's sine-principle

Author: Andrey Bovykin
Journal: Proc. Amer. Math. Soc. 135 (2007), 2967-2973
MSC (2000): Primary 03F30, 03F99; Secondary 05D10
Published electronically: May 8, 2007
MathSciNet review: 2317975
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Abstract: For every $ n\geq 1$ and every function $ F$ of one argument, we introduce the statement $ \SP_F^n$: ``for all $ m$, there is $ N$ such that for any set $ A=\{a_1, a_2, \ldots, a_N\}$ of rational numbers, there is $ H\subseteq A$ of size $ m$ such that for any two $ n$-element subsets $ a_{i_1}< a_{i_2}<\cdots < a_{i_n}$ and $ a_{i_1}< a_{k_2}< \cdots < a_{k_n}$ in $ H$, we have

$\displaystyle \vert\sin(a_{i_1}\cdot a_{i_2} \cdots a_{i_n}) - \sin(a_{i_1}\cdot a_{k_2} \cdots a_{k_n} )\vert< F(i_1)''. $

We prove that for $ n\geq 2$ and any function $ F(x)$ eventually dominated by $ ({2 \over 3})^{\log^{(n-1)}(x)}$, the principle $ \SP_F^{n+1}$ is not provable in $ I\Sigma_n$. In particular, the statement $ \forall n \SP_{({2 \over 3})^{\log^{(n-1)}}}^n$ is not provable in Peano Arithmetic. In dimension 2, the result is: $ I\Sigma_1$ does not prove $ \SP^2_F$, where $ F(x)=({2 \over 3})^{\sqrt[A^{-1}(x)]{x}}$ and $ A^{-1}$ is the inverse of the Ackermann function.

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Additional Information

Andrey Bovykin
Affiliation: Steklov Mathematical Institute, Fontanka 27, St. Petersburg, Russia; Liverpool University, Liverpool, United Kingdom

Keywords: Unprovable combinatorial statements, irrationality measure of $\pi$, dynamical system, Paris-Harrington Principle, Kanamori-McAloon Principle.
Received by editor(s): June 7, 2006
Published electronically: May 8, 2007
Communicated by: Julia Knight
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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