1 · The essentials
The manuscript studies a family of probability distributions called unbiased Bernoulli convolutions, each controlled by a number between zero and one. It asks which distributions have a density and which instead concentrate on a set of zero length. The manuscript claims a classification covering every parameter in this interval. Its rule depends on how the parameter is approached from one side by numbers drawn from explicitly defined finite collections. Although each collection is finite, the rule involves an infinite approximation condition, not a finite membership test. The manuscript also claims examples of concentration beyond the reciprocal Pisot parameters singled out in the source. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine checking a location against an endless sequence of maps, each carrying only finitely many marked landmarks. The question is whether landmarks approach that location from the required side in the required way; consulting just one map is not the same as checking the whole condition.
Why it matters
The claimed classification connects the shape of these probability distributions with arithmetic approximation. Its scope is every parameter in , rather than only selected examples. The additional singularity claims extend the examples beyond reciprocals of Pisot numbers.
2 · Going further
The precise classification problem concerns unbiased Bernoulli convolutions with parameter : determine which are singular and which are absolutely continuous. Absolute continuity means that the distribution has a density relative to ordinary length; singularity means that its probability is concentrated on a set of zero length. The abstract claims an arithmetic criterion distinguishing these behaviors throughout the stated interval.
The mechanism described in the abstract is one-sided approximation using explicitly defined finite sets of algebraic units. The classification depends on an infinite approximation condition. Consequently, the finiteness of the individual sets should not be confused with a finite membership algorithm. The supplied abstract does not give the sets, the approximation inequalities, or the steps of the argument, so a more detailed account of the method is not available from this source.
The manuscript also claims singularity at for every quartic Salem number . It further claims singularity at the reciprocal of a specified non-Pisot root of an explicit degree-31 polynomial. These are presented as singular examples beyond reciprocal Pisot parameters; neither the polynomial nor its specified root is reproduced in the supplied material. Only the main manuscript is supplied, so no relationship to companion manuscripts can be described.
Useful notions
- Probability distributions and densities
- Sets of zero length
- Approximation from one side
- Algebraic numbers and algebraic units
- Finite tests versus infinite conditions
What was known before / what is new
Before
The source uses reciprocals of Pisot numbers as the reference class beyond which the manuscript supplies singular examples. It does not provide a historical account of earlier classification results or identify a previously posed open problem.
New
The manuscript claims a classification for every through infinite, one-sided approximation by explicit finite sets of algebraic units. It also claims singularity at reciprocals of every quartic Salem number in and at the reciprocal of a specified non-Pisot root of an explicit degree-31 polynomial.
Limits and assumptions
The classification concerns unbiased Bernoulli convolutions and parameters strictly between zero and one. The Salem-number statement is restricted to quartic Salem numbers in , while the degree-31 statement concerns a specified root, not arbitrary non-Pisot numbers. The arithmetic criterion is an infinite approximation condition, not a finite membership algorithm. The supplied abstract omits the exact criterion and argument details, and it states no remaining open questions. The main result is not formalized in Lean and has not been formally verified.
Mini glossary
- Unbiased Bernoulli convolutions —
- The parameter-dependent family of probability distributions classified in the manuscript.
- Absolute continuity —
- The property of having a probability density relative to ordinary length.
- Singularity —
- The property of concentrating all probability on a set of zero length.
- One-sided approximation —
- Approximation in which the approximating numbers approach the target from a specified side.
- Finite membership algorithm —
- A finite procedure deciding whether a parameter belongs to a specified class, which the abstract explicitly distinguishes from its infinite approximation criterion.
Keywords : Bernoulli convolutions · Arithmetic approximation · Singularity
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
Arithmetic classification and non-Pisot singularity for Bernoulli convolutions
03/10/2026We 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.