Taking the Convoluted Out of Bernoulli Convolutions: A Discrete Approach

Abstract

Abstract. In this paper we consider a discrete version of the Bernoulli convolution problem traditionally studied via functional analysis. We discuss several innovative algorithms for computing the sequences with this new approach. In particular, these algorithms assist us in gathering data regarding the maximum values. By looking at a family of associated polynomials, we gain insight on the local behavior of the sequence itself. This work was completed as part of the Clemson University REU, an NSF funded program 1 .

Discrete Bernoulli convolutions: An algorithmic approach toward bound improvement

“…We ran many jobs using the high-throughput computing facilities of Condor at Clemson University. Before implementing this algorithm, the best known bound on m n was O(( √ 2) n ) [2]. We succeeded in significantly improving the bound.…”

“…We are able to choose exactly how we break up the interval [1/3, 1/2] in (2.3), and we choose the pullback number r defined in step (2). There is a computational trade-off when setting these two values-it is hard to make both large at once.…”

“…Figure 1 shows the first few levels of the Bernoulli sequence. In this paper, we are interested in the rate at which the maximum m n := max(B n ) is growing with n. It is easy to see that the mean μ(B n ) = (4/3) n [2]. The major question to be asked is if m n also grows like (4/3) n .…”

“…Note that F q (t) is absolutely continuous at q = 2/3 if and only if m n = O((4/3) n ). For a detailed introduction and motivation on the discrete analogue of Bernoulli convolutions, see [2].…”

Bernoulli convolutions and 1D dynamics

“…For the moment however, our results on piecewise convexity are restricted to some special values of λ, although we are able to run numerical approximations for any λ. There have been previous numerical investigations into Bernoulli convolutions, we mention in particular the work of Benjamini and Solomyak [1] and of Calkin et al [2,3].…”