OpenAI · 153-01 · 2026-10-03 · Dynamical systems and ergodic theory
Arithmetic classification and non-Pisot singularity for Bernoulli convolutions
preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026
Family 153. Arithmetic classification of Bernoulli convolutions.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
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Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every by an infinite, one-sided approximation condition using explicit finite sets of algebraic units. It also proves singularity at reciprocals of every quartic Salem number in , giving examples beyond reciprocal Pisot parameters.
We give an arithmetic classification of singular and absolutely continuous unbiased Bernoulli convolutions for every parameter . 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 and at the reciprocal of a specified non-Pisot root of an explicit degree-31 polynomial.
Preprint source and Lean scope
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