Let \beta\in(1,2) be a Salem number of degree four. Then the unbiased Bernoulli convolution \nu_{1/\beta} is singular with respect to Lebesgue measure.
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3 October 2026, current edition.
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We give an arithmetic classification of singular and absolutely continuous unbiased Bernoulli convolutions for every parameter \(\lambda\in(0,1)\). The criterion is expressed through one-sided approximation by explicitly defined finite sets of algebraic units; it is an infinite approximation condition, not a finite membership algorithm. We also establish singular examples beyond reciprocals of Pisot numbers: singularity holds at the reciprocal of every quartic Salem number in \((1,2)\) and at the reciprocal of a specified non-Pisot root of an explicit degree-31 polynomial.
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Author not stated. Source
From terrytao.wordpress.com, 8 October 2026. Source
Joerg Neunhaeuserer says they spent much of their career on a problem now claimed by OpenAI (they cite the number 153; read here as the family on Bernoulli convolutions), is relieved not to have to do that work, and is proud their intuition about the answer was right.
Guozheng Cheng, Xiang Fang, Xueqing Ma, Hongli Zhang, Strict correlation-dimension drop for quartic Salem Bernoulli convolutions, 7 October 2026. Source
Cheng, Fang, Ma and Zhang say OpenAI’s Bernoulli-convolution manuscript proves singularity for every quartic Salem number in (1,2) by a different argument and does not imply their strict correlation-dimension drop.
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