Dimension of images of large level sets - Armstrong, Gavin; O’Farrell, Anthony G. (MATHEMATICA SCANDINAVICA, 2022)
Let k be a natural number. We consider k-times continuously differentiable real-valued functions fE→ R , where E is some interval on the line having positive length. For 0<α <1 let I_α (f) denote the set of values yR whose preimage f^-1(y) has Hausdorff dimension at least α. We consider how large can be the Hausdorff dimension of I_α (f), as f ranges over the set of all k-times continuously differentiable functions from E into R . We show that the sharp upper bound on I_α (f) is (1-α )/k.