We consider Poisson Boolean percolation on \mathbb R^d with power-law distribution on the radius with a finite d-moment for d\ge 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 where subcritical sharpness is proved under the assumption that the radii distribution has a 5d-3 finite moment. Our proofs techniques are different from their paper: we do not use 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.
