Singular Bernoulli convolutions at non-Pisot parameters

Two non-Pisot singularity theorems. If $\beta\in(1,2)$ is a Salem number of degree four then $\nu_{1/\beta}$ is singular; explicit cases are the real roots of $x^4-x^3-x^2-x+1$ and $x^4-2x^3+x^2-2x+1$. An explicit degree-31 polynomial has a unique real root $\beta\in(1.8392,1.8393)$, not Pisot, with $\nu_{1/\beta}$ singular. The manuscript also proves a criterion: $\nu_\lambda$ is singular iff $\lambda$ lies in an explicit limsup set of one-sided approximations by finite sets of algebraic units; the paper says this is an infinite approximation condition, not a finite membership test, so it is not an effective classification. It does not decide Salem numbers of degree six or more, nor which non-Pisot parameters are singular in general.

Two non-Pisot singularity theorems. If $\beta\in(1,2)$ is a Salem number of degree four then $\nu_{1/\beta}$ is singular; explicit cases are the real roots of $x^4-x^3-x^2-x+1$ and $x^4-2x^3+x^2-2x+1$. An explicit degree-31 polynomial has a unique real root $\beta\in(1.8392,1.8393)$, not Pisot, with $\nu_{1/\beta}$ singular. The manuscript also proves a criterion: $\nu_\lambda$ is singular iff $\lambda$ lies in an explicit limsup set of one-sided approximations by finite sets of algebraic units; the paper says this is an infinite approximation condition, not a finite membership test, so it is not an effective classification. It does not decide Salem numbers of degree six or more, nor which non-Pisot parameters are singular in general.