Asymptotics of signed Bernoulli convolutions scaled by multinacci numbers

TL;DR

This paper investigates the asymptotic behavior of signed Bernoulli convolutions scaled by multinacci numbers, revealing a coincidence with unsigned convolutions for odd degrees and providing asymptotic results for even degrees.

Contribution

It establishes the exact asymptotics of signed Bernoulli convolutions for multinacci numbers, highlighting differences based on the parity of the defining integer m.

Findings

For odd m, the variation of signed convolutions equals the unsigned convolution.

For even m, the paper derives the precise asymptotic behavior of the total variation.

The results connect properties of Bernoulli convolutions with algebraic properties of multinacci numbers.

Abstract

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 some integer $m\ge 2$. 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 $m$ is even, we obtain the exact asymptotic of the total variation $\|\nu_\beta^{(n)}\|$ as $n\rightarrow\infty$.