{"id":"openai-math/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions","pdf":"https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026","tags":["bernoulli-convolutions","number-theory","fractals"],"score":0,"title":"Arithmetic classification and non-Pisot singularity for Bernoulli convolutions","domain":"Dynamical systems and ergodic theory","family":"153","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DS","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-03","reactions":{},"formalized":false,"family_title":"Arithmetic classification and non-Pisot singularity for Bernoulli convolutions"}