Singular non-Pisot Bernoulli convolutions

TL;DR

This paper identifies specific algebraic numbers, including Salem and non-Pisot numbers, for which Bernoulli convolutions are singular, providing new explicit examples after decades of research.

Contribution

It introduces the first explicit new examples of singular Bernoulli convolutions involving Salem and non-Pisot algebraic numbers since 1939.

Findings

Found two countable collections of Salem numbers in (1,2)

Identified a Salem number and a non-Pisot, non-Salem algebraic number in (1,2)

Discovered a non-Pisot, non-Salem algebraic number greater than 3

Abstract

We identify a family of numbers for which the Bernoulli convolution is singular. Within this family we find two countable collections of Salem numbers in the interval $(1,2)$, and another Salem number and an algebraic integer that is neither Pisot nor Salem in $(1,2)$. It also contains a non-Pisot, non-Salem algebraic number bigger than 3. Hence, we provide the first new explicit examples of singular Bernoulli convolutions since the work of Erd\H{o}s in 1939.