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We study the signed Bernoulli convolution $$\nu_\beta^{(n)}=*_{j=1}^n \left(\frac12\delta_{\beta^{-j}}-\frac12\delta_{-\beta^{-j}}\right ),\ n\ge 1$$where $\beta>1$ satisfies $$\beta^m=\beta^{m-1}+\cdots+\beta+1$$ for someinteger $m\ge 2$. When $m$ is odd, we show that the variation$|\nu_\beta^{(n)}|$ coincides the unsigned Bernoulli convolution$$\mu_\beta^{(n)}=*_{j=1}^n \left(\frac12\delta_{\beta^{-j}}+\frac12\delta_{-\beta^{-j}}\right ).$$ When $m$ iseven, we obtain the exact asymptotic of the total variation$\|\nu_\beta^{(n)}\|$ as $n\rightarrow\infty$.

Asymptotics of signed Bernoulli convolutions scaled by multinacci numbers