Smoothness of random self-similar measures on the line and the existence of interior points

• 成果类型: Article; Early Access
• 署名作者: Barany, Balazs; Rams, Michal
• 署名单位: Budapest University of Technology & Economics; Polish Academy of Sciences; Institute of Mathematics of the Polish Academy of Sciences
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01389-2
• 发表日期: 2025
• 关键词: absolute continuity; similar sets; dimension

摘要

In this paper, we study the smoothness of the density function of absolutely continuous measures supported on random self-similar sets on the line. We show that the natural projection of a measure with symbolic local dimension greater than 1 at every point is absolutely continuous with H & ouml;lder continuous density almost surely. In particular, if the similarity dimension is greater than 1 then the random self-similar set on the line contains an interior point almost surely.