# FOM: 152:sin

friedman@math.ohio-state.edu friedman at math.ohio-state.edu
Sat Jun 8 22:48:24 EDT 2002

```THEOREM 1. Let n >> k and x_1,...,x_n be real numbers. There exist p_1 < ... <
p_k+2 such that
|sin(x_p_1 x_p_2 ... x_p_k) - sin(x_p_1 x_p_3 ... x_p_k+1)| < 4^-p_1;
|sin(x_p_2 x_p_3 ... x_p_k+1) - sin(x_p_2 x_p_4 ... x_p_k+2)| < 4^-p_2.

We can write the latter as a finite statement using the rationals.

THEOREM 2. Let n >> k and x_1,...,x_n be rational numbers. There exist p_1 <
.. < p_k+2 such that
|sin(x_p_1 x_p_2 ... x_p_k) - sin(x_p_1 x_p_3 ... x_p_k+1)| < 4^-p_1;
|sin(x_p_2 x_p_3 ... x_p_k+1) - sin(x_p_2 x_p_4 ... x_p_k+2)| < 4^-p_2.

THEOREM 3. Theorem 2 is provable in ACA but not in Peano arithmetic. The best n
= n(k) of both Theorems is epsilon_0-recursive but eventually dominates each <
epsilon_0-recursive function.

NOTE: We did not attempt to give the sharpest results.

RECOMMENDED: 126, 150, 151.

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This is the 152nd in a series of self contained postings to FOM covering
a wide range of topics in f.o.m. Previous ones counting from #100 are:

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102: Turing Degrees/2  4/8/01  5:20PM
103:Hilbert's Program for Consistency Proofs/1 4/11/01  11:10AM
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108:Finite Boolean Relation Theory   9/18/01  12:20PM
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114:Borel Functions on HC  1/1/02  1:38PM
115:Aspects of Coloring Integers  1/3/02  10:02PM
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118:Discrepancy Theory/2   1/20/02  1:31PM
119:Discrepancy Theory/3  1/22.02  5:27PM
120:Discrepancy Theory/4  1/26/02  1:33PM
121:Discrepancy Theory/4-revised  1/31/02  11:34AM
122:Communicating Minds IV-revised  1/31/02  2:48PM
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