Rational approximations of real numbers

For anyx ∈ r put $$c(x) = overline {mathop {lim }limits_{t to infty } } mathop {min }limits_{(p,qmathop {) in Z}limits_{q leqslant t} times N} tleft| {qx - p} right|.$$ . Let [x₀; x₁,..., x_n, ...] be an expansion of x into a continued fraction and let (M = { x in J,overline {mathop {lim }limits_{n to infty } } x_n< infty }) .Forx∈M put D(x)=c(x)/(1?c(x)). The structure of the set (mathfrak{D} = { D(x),x in M}) is studied. It is shown that $$mathfrak{D} cap (3 + sqrt 3 ,(5 + 3sqrt 3 )/2) = { D(x^{(n,3} )} _{n = 0}^infty nearrow (5 + 3sqrt 3 )/2,$$ where (x^{(n,3)} = [overline {3;(1,2)_n ,1} ].) This yields for (mu = inf { z,mathfrak{D} supset (z, + infty )}) (“origin of the ray”) the following lower bound: μ?(5+3√3)/2=5.0n>(5 + 3/3)/2=5.098.... Suppose a∈n. Put (M(a) = { x in M,overline {mathop {lim }limits_{n to infty } } x_n = a}) , (mathfrak{D}(a) = { D(x),x in M(a)}) . The smallest limit point of (mathfrak{D}(a)(a geqslant 2)) is found. The structure of (a) is studied completely up to the smallest limit point and elucidated to the right of it.