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:
100:Boolean Relation Theory IV corrected 3/21/01 11:29AM
101:Turing Degrees/1 4/2/01 3:32AM
102: Turing Degrees/2 4/8/01 5:20PM
103:Hilbert's Program for Consistency Proofs/1 4/11/01 11:10AM
104:Turing Degrees/3 4/12/01 3:19PM
105:Turing Degrees/4 4/26/01 7:44PM
106.Degenerative Cloning 5/4/01 10:57AM
107:Automated Proof Checking 5/25/01 4:32AM
108:Finite Boolean Relation Theory 9/18/01 12:20PM
109:Natural Nonrecursive Sets 9/26/01 4:41PM
110:Communicating Minds I 12/19/01 1:27PM
111:Communicating Minds II 12/22/01 8:28AM
112:Communicating MInds III 12/23/01 8:11PM
113:Coloring Integers 12/31/01 12:42PM
114:Borel Functions on HC 1/1/02 1:38PM
115:Aspects of Coloring Integers 1/3/02 10:02PM
116:Communicating Minds IV 1/4/02 2:02AM
117:Discrepancy Theory 1/6/02 12:53AM
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
123:Divisibility 2/2/02 10:57PM
124:Disjoint Unions 2/18/02 7:51AM
125:Disjoint Unions/First Classifications 3/1/02 6:19AM
126:Correction 3/9/02 2:10AM
127:Combinatorial conditions/BRT 3/11/02 3:34AM
128:Finite BRT/Collapsing Triples 3/11/02 3:34AM
129:Finite BRT/Improvements 3/20/02 12:48AM
130:Finite BRT/More 3/21/02 4:32AM
131:Finite BRT/More/Correction 3/21/02 5:39PM
132: Finite BRT/cleaner 3/25/02 12:08AM
133:BRT/polynomials/affine maps 3/25/02 12:08AM
134:BRT/summation/polynomials 3/26/02 7:26PM
135:BRT/A Delta fA/A U. fA 3/27/02 5:45PM
136:BRT/A Delta fA/A U. fA/nicer 3/28/02 1:47AM
137:BRT/A Delta fA/A U. fA/beautification 3/28/02 4:30PM
138:BRT/A Delta fA/A U. fA/more beautification 3/28/02 5:35PM
139:BRT/A Delta fA/A U. fA/better 3/28/02 10:07PM
140:BRT/A Delta fA/A U. fA/yet better 3/29/02 10:12PM
141:BRT/A Delta fA/A U. fA/grammatical improvement 3/29/02 10:43PM
142:BRT/A Delta fA/A U. fA/progress 3/30/02 8:47PM
143:BRT/A Delta fA/A U. fA/major overhaul 5/2/02 2:22PM
144: BRT/A Delta fA/A U. fA/finesse 4/3/02 4:29AM
145:BRT/A U. B U. TB/simplification/new chapter 4/4/02 4:01AM
146:Large large cardinals 4/18/02 4:30AM
147:Another Way 7:21AM 4/22/02
148:Finite forms by relativization 2:55AM 5/15/02
149:Bad Typo 1:59PM 5/15/02
150:Finite obstruction/statisics 8:55AM 6/1/02
151:Finite forms by bounding 4:35AM 6/5/02
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