Almost sharp sharpness for Poisson Boolean percolation

We consider Poisson Boolean percolation on Rd with power-law distribution on the radius with a finite d-moment for d≥2. We prove that subcritical sharpness occurs for all but a countable number of power-law distributions. This extends the results of Duminil-Copin–Raoufi–Tassion (Ann. Henri Lebesgue 3 (2020) 677–700) where subcritical sharpness is proved under the assumption that the radii distribution has a 5d−3 finite moment. Our proofs techniques are different from (Ann. Henri Lebesgue 3 (2020) 677–700): we do not use a randomized algorithm and rely on specific independence properties of Boolean percolation, inherited from the underlying Poisson process. We also prove supercritical sharpness for any distribution with a finite d-moment and the continuity of the critical parameter for the truncated distribution when the truncation goes to infinity.