广义Besicovitch集的Hausdorff测度

The Hausdorff measure of generalized Besicovitch set

给定一概率向量p=(p0,p1,…,pm-1)(m≥2),Besicovitch集Bp是由单位区间[0,1]中那些在m-进制展式中j(j=0,1,…,m-1)出现的频率为pj的点组成,即Bp={x∈[0,1]:limn→∞1n∑nk=1τj(xk)=pj,j=0,1,…,m-1},其中τj(·)表示单点集{j}的特征函数.对给定的概率向量p=(p0,p1,…,pm-1)以及满足一定条件的实值向量a=(a0,a1,…,am-1),考虑广义Besicovitch集Bτ,a={x∈[0,1]:}limn→∞1nτ(∑nk=1τj(xk)-npj)=aj,j=0,1,…,m-1},其中τ∈(12,1),并证明了Bτ,a在任何量纲函数下的Hausdorff测度非零即无穷大,进一步,证明了当∑m-1j=0ajlogpj≤0时,广义Besicovitch集的Hausdorff测度为无穷大.

For a given probability vector p=(p_0,p_1,...,p_(m-1))(m≥2),Besicovitch set B_p consists of the points in the unit interval I=of which the frequency of j(j=0,1,...,m-1) is p_j in the m-expansion.That is,B_p={x∈:(lim)n→∞1n∑nk=1τ_j(x_k)=p_j,j=0,1,...,m-1},where τ_j(·) is the indicator function of the set {j}.For a given probability p=(p_0,p_1,...,p_(m-1)) and a real-valued vetor a=(a_0,a_1,...,a_(m-1)) which satisfying certain conditions,consider generalized Besicovitch set B_(τ,a)={x∈:}(lim)n→∞1n~τ(∑nk=1τ_j(x_k)-np_j)=a_j,j=0,1,...,m-1},where τ∈(12,1),proved that the Hausdorff measure of B_(τ,a) is either zero or infinity for any gauge functions.And furthermore,got that the Hausdorff measure of B_(τ,a) is infinity when ∑m-1j=0a_jlogp_j≤0.