Unprovability of sharp versions of Friedman’s sine-principle

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For every n ≥ 1 n\geq 1 and every function F F of one argument, we introduce the statement S P F n \mathrm {SP}_F^n : “for all m m , there is N N such that for any set A = { a 1 , a 2 , … , a N } A=\{a_1, a_2, \ldots , a_N\} of rational numbers, there is H ⊆ A H\subseteq A of size m m such that for any two n n -element subsets a i 1 > a i 2 > ⋯ > a i n a_{i_1}> a_{i_2}>\cdots > a_{i_n} and a i 1 > a k 2 > ⋯ > a k n a_{i_1}> a_{k_2}> \cdots > a_{k_n} in H H , we have \[ | sin ⁡ ( a i 1 ⋅ a i 2 ⋯ a i n ) − sin ⁡ ( a i 1 ⋅ a k 2 ⋯ a k n ) | > F ( i 1 ) " . |\sin (a_{i_1}\cdot a_{i_2} \cdots a_{i_n}) - \sin (a_{i_1}\cdot a_{k_2} \cdots a_{k_n} )|> F(i_1)". \] We prove that for n ≥ 2 n\geq 2 and any function F ( x ) F(x) eventually dominated by ( 2 3 ) log ( n − 1 ) ⁡ ( x ) ({2 \over 3})^{\log ^{(n-1)}(x)} , the principle S P F n + 1 \mathrm {SP}_F^{n+1} is not provable in I Σ n I\Sigma _n . In particular, the statement ∀ n S P ( 2 3 ) log ( n − 1 ) n \forall n \mathrm {SP}_{({2 \over 3})^{\log ^{(n-1)}}}^n is not provable in Peano Arithmetic. In dimension 2, the result is: I Σ 1 I\Sigma _1 does not prove S P F 2 \mathrm {SP}^2_F , where F ( x ) = ( 2 3 ) x A − 1 ( x ) F(x)=({2 \over 3})^{\sqrt [A^{-1}(x)]{x}} and A − 1 A^{-1} is the inverse of the Ackermann function.