The Hausdorff Dimension Of The Cantor Sets Arising From Continued Radicals

If a1, a2, a3, … are nonnegative real numbers and , then is a continued radical with terms a1 a2, a3, … The set of real numbers representable as a continued radical whose terms ai are all from s set S={a, b) of two natural numbers is a cantor. T. Clark·T. Richmond has investigated the thickness,measure, and sums of such Cantor sets.The article tries to investigate the Hausdorff dimension of them.T. Clark·T. Richmond[3] has calculated the thickness ?(a,b) of cantor sets C({a,b}),and ? (a,b) can be written as a limit of continued radicals whose terms from a or b.In [29],the relation between the Hausdorff dimension of a cantor set C ({a, b}) and its thickness ?(a,b) is:It is the lower bound of the Hausdorff dimension of C({a, b}).T. Clark-T. Richmond[3] found that the longest bridge on level n is the left-most bridge.2" intervals with the length of the measure of the left-most bridge can cover the n-order basic interval of C({a, b}).According to some corresponding techniques for the length,we can have an expression about n.Finding a constant as the exponents of the expression,the product of 2" and the expression to the power of the constant is a constant. So,the exponents is an upper bound of the Hausdorff dimension of C({a,b}).The article tries to make it clear that what is the lower bound and the upper bound of the Hausdorff dimension of C({a, b}).About the accuracy value for the Hausdorff dimension of the cantor set C({a,b}),we can give the elegant formula in terms of pressureSo,the unique real solution of the equation P(-slog?f'?)=0 is the Hausdorff dimension of it.