Strict correlation-dimension drop for quartic Salem Bernoulli convolutions

Abstract

For every quartic Salem number $\beta \in (1,2)$, we prove that the Bernoulli convolution ${\nu }_{{\beta }^{-1}}$ with equal weights has correlation dimension strictly less than one. In particular, neither of the two corresponding measures has an $L^2$ density. We relate correlation dimension to exact collision probabilities and express these probabilities as positive integrals over a four-dimensional torus associated with the defining polynomial. The unit-circle conjugates produce a persistent oscillatory term; using carefully chosen return times of the corresponding rotation, we take differences that make this term small while controlling the length and cost of the resulting configurations. Many well-separated placements of these configurations then give a collision probability above the critical scale, which yields the strict dimension drop. The qualitative argument is independent of the computer-assisted estimates. Separate computer-assisted arguments give \[ 0.99999<D_2(ν_{β_1^{-1}})<1-10^{-65}, \qquad 0.99999<D_2(ν_{β_2^{-1}})<1-10^{-75}, \] where ${\beta }_1<{\beta }_2$ are the two quartic Salem numbers in $(1,2)$.