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Bernoulli Convolution

An unbiased Bernoulli convolution nu_lambda, for 0<lambda<1, is the distribution of the random series

Y_lambda=sum_(n=0)^inftyepsilon_nlambda^n,

where the epsilon_n are independent random variables taking each value -1 and 1 with probability 1/2. It is a self-similar measure for the two maps x|->lambdax-1 and x|->lambdax+1, with equal weights. For lambda<1/2, its support has Lebesgue measure zero. For lambda=1/2, it is the uniform distribution on [-2,2].

Cheng et al. (2026) proved that dim_Hnu_(1/beta)<1 for every quartic Salem constant beta. Here dim_H denotes Hausdorff dimension of a measure, the infimum of the Hausdorff dimensions of measurable sets having full measure. This strengthens singularity by giving a strict dimension drop. Cheng et al. (2026) report GPT-6.0 Sol assistance with local mathematical reasoning and numerical checks, with the strategy supplied by the authors.

See also

Bernoulli Distribution, Lebesgue Measure, Pisot Number, Salem Constants, Self-Similar Measure, Singular Measure

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References

Cheng, G.; Fang, X.; Ma, X.; and Zhang, H. "Strict Hausdorff-Dimension Drop for Quartic Salem Bernoulli Convolutions." 8 Oct 2026. https://arxiv.org/abs/2610.12072.

Cite this as:

Weisstein, Eric W. "Bernoulli Convolution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BernoulliConvolution.html

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