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Answering two questions of Beresnevich and Velani, we develop zero‐one laws in both simultaneous and multiplicative Diophantine approximation. Our proofs mainly rely on a versatile Cassels–Gallagher type theorem and the cross fibering principle of Beresnevich, Haynes and Velani.

ZERO‐ONE LAWS IN SIMULTANEOUS AND MULTIPLICATIVE DIOPHANTINE APPROXIMATION