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Consider an inviscid Burgers equation whose initial data is a Levy a-stableprocess Z with a > 1. We show that when Z has positive jumps, the Hausdorffdimension of the set of Lagrangian regular points associated with the equationis strictly smaller than 1/a, as soon as a is close to 1. This gives a negativeanswer to a conjecture of Janicki and Woyczynski. Along the way, we contradicta recent conjecture of Z. Shi about the lower tails of integrated stableprocesses

On the Hausdorff Dimension of Regular Points of Inviscid Burgers Equation with Stable Initial Data