Hausdorff dimension of deranged Cantor set without some boundedness condition

Abstract

A deranged Cantor set(without the uniform boundedness conditionaway from zero of contraction ratios) whose weak local dimensionsfor all points coincide has its Hausdorff dimension of the samevalue of weak local dimension. We will show it using an energytheory instead of Frostman's density lemma which was used for thecase of the deranged Cantor set with the uniform boundednesscondition of contraction ratios. In the end, we will give anexample of such a deranged Cantor set.