A NULLSTELLENSATZ FOR AMOEBAS

DISTORTION OF HAUSDORFF MEASURES AND REMOVABILITY 569
In the above inductive construction, we can then choose the σ j ’s so that
v(σ1
K ···σ N K) ≤ 2−N(1−1/K) for every index N. Now,(5.4)and(5.8) imply that
ε ′ (t N ) 2 ≤ ε(s N ) 1+1/K 2 N(1−1/K) ≤ ε(s N) 1+1/K
.
v(s N )
On the other hand, by (5.5), we also have t N−1 /t N ≤ s N−1 /s N , and so we may extend
ε ′ (t), determined by (5.8) only at the t N ’s, so that it is continuous, nondecreasing, and
satisfies
∫
0
ε ′ (t) 2 dt
t
∫
≤
0
ε(s) 1+1/K
v(s)
ds
s < ∞.
Hence the claim follows. Combining it with Mattila’s theorem [21, Theorem 3.8]
completes the proof of the theorem.
Lastly, let us note that if we do not care for the analytic capacity of the target set, a
straightforward modification of Theorem 5.1, normalizing the disks of the construction
so that m N t N η(t N ) = 1, gives the following.
COROLLARY 5.2
Let K ≥ 1, and let h(t) = tη(t) be a measure function such that
• η is continuous and nondecreasing, η(0) = 0, and η(t) = 1 whenever t ≥ 1;
• lim
t→0
(t α /η(t)) = 0 for all α>0.
There exist a compact set E ⊂ D and a K-quasiconformal mapping φ such that
dim(E) = 2
K + 1
and H h( φ(E) ) = 1. (5.11)
Note added in proof. In a recent work, Bishop [10] has given a negative answer to
Question 2.4. However, Conjecture 2.3 remains open.
On the other hand, Uriarte-Tuero [31] has recently given a positive answer to
Question 4.2.
References
[1] D. R. ADAMS and L. I. HEDBERG, Function Spaces and Potential Theory, Grundlehren
Math. Wiss. 314, Springer, Berlin, 1996. MR 1411441 551, 553
[2] L. V. AHLFORS, Bounded analytic functions, Duke Math. J. 14 (1947), 1 – 11.
MR 0021108 541