Null sets for doubling and dyadic doubling measures

In this note, we study sets on the real line which are null with respect to all doubling measures on R , or with respect to all dyadic doubling measures on R . We give some sufficient conditions for the former, a test for the latter, and some examples. Our work is motivated by a characterization of dyadic doubling measures by Fefferman, Kenig and Pipher [5], and by a result of Martio [8] on porous sets and sets of total A -harmonic measure zero for certain class of nonlinear A -operators. A measure μ on R is said to have the doubling property with constant λ if, whenever I and J are two neighboring intervals of same length then μ(I) ≤ λμ(J) ; denote by D(λ) the collection of all doubling measures with constant λ , and D = ∪λ≥1D(λ) . A measure μ on R has the dyadic doubling property with constant λ if μ(I) ≤ λμ(J) whenever I and J are two dyadic neighboring intervals of same length and I ∪ J is also a dyadic interval; denote by Dd(λ) and Dd the corresponding collections of dyadic doubling measures. Given {an} , 0 0. Theorem 1. If 0 1. In Theorem 3, we give a deterministic procedure of testing whether E is Dd(λ)-null. In this process, an optimal measure μE,λ among Dd(λ) is selected for E . The precise statement is given in Section 2. Denote by N the collection of null sets for doubling measures { E : μ(E) = 0 for all μ ∈ D } , and Nd its dyadic counterpart { E : μ(E) = 0 for all μ ∈ Dd } . Clearly Nd ⊆ N and N is translation invariant. The assertion that Nd 6= N is not suprising, however it requires a lot of work. Theorem 4. There exists a perfect set S ⊆ [0, 1] which is in N \ Nd . And corresponding to this S , there exists a set T of dimension one, so that t+S ∈ Nd for each t ∈ T . It would be interesting to know whether a pair of sets S , T can be chosen to satisfy length (T ) > 0 in addition to the properties in Theorem 4. Theorem 5. Let t be any number whose binary expansion has infinitely many zeros and infinitely many ones. Then there exists a perfect set St so that St ∈ N \ Nd but t+ St ∈ Nd . Finally, in Section 4, we shall comment on relations betwen sets in N and null sets of the harmonic measures with respect to the p-Laplacians in the upper half plane. The author would like to thank A. Hinkkanen for his joint contribution on Theorem 1, and R. Kaufman for many conversations. 1. Proofs of Theorems 1 and 2 We first state a useful lemma. Lemma 1. Let μ be a dyadic doubling measure on [0, 1] and I be any subinterval. Then there exists K > 1 depending on the dyadic doubling constant λ only, so that 4|I|μ ( [0, 1] ) ≥ μ(I) ≥ 1 4 |I|μ (