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Lower bounds on Bourgain’s constant for harmonic measure
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 27 October 2023, pp. 1-20
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For every
$n\geq 2$, Bourgain’s constant 
$b_n$ is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most 
$n-b_n$ for every domain in 
$\mathbb {R}^n$ on which harmonic measure is defined. Jones and Wolff (1988, Acta Mathematica 161, 131–144) proved that 
$b_2=1$. When 
$n\geq 3$, Bourgain (1987, Inventiones Mathematicae 87, 477–483) proved that 
$b_n>0$ and Wolff (1995, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton University Press, Princeton, NJ, 321–384) produced examples showing 
$b_n<1$. Refining Bourgain’s original outline, we prove that for all
$$\begin{align*}b_n\geq c\,n^{-2n(n-1)}/\ln(n),\end{align*}$$
$n\geq 3$, where 
$c>0$ is a constant that is independent of n. We further estimate 
$b_3\geq 1\times 10^{-15}$ and 
$b_4\geq 2\times 10^{-26}$.
