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It is well known that standard one-dimensional Brownian motion B(t) has no isolated zeros almost surely. We show that for any α < 1/2 there are α-Hölder continuous functions f for which the process B − f has isolated zeros with positive probability. We also prove that for any continuous function f , the zero set of B − f has Hausdorff dimension at least 1/2 with positive probability, and 1/2 is an upper bound on the Hausdorff dimension if f is 1/2-Hölder continuous or of bounded variation .doi:10.1214/ejp.v16-927 fatcat:36ihd24uqrah3auu4qp4sxesja