Hausdorff dimension of symmetric perfect sets

`We consider nowhere dense perfect subsets of [0,1] that are symmetric but have no additional nice properties. We prove that if E = boolean AND E-n is a symmetric perfect set and the length of the basic intervals in E-n is denoted by I-n then the Hausdorff dimension of E is s = lim (n-->infinity) inf{s(n): 2(n)l(n)(sn) = 1} = lim (n-->infinity) inf log 2(n) / -log l(n). The argument we use also shows that using natural covers of E; i.e., covers consisting of the 2(n) closed, equal length intervals of the nth stage, yield an estimate for the s-dimensional Hausdorff measure within a factor of four.