Abstract
Suppose \({\Lambda}\) is a discrete infinite set of nonnegative real numbers. We say that \({\Lambda}\) is type 1 if the series \(s(x)=\sum\nolimits_{\lambda\in\Lambda}f(x+\lambda)\) satisfies a “zero-one” law. This means that for any non-negative measurable \(f \colon \mathbb{R} \to [0,+ {\infty})\) either the convergence set \(C(f, {\Lambda})=\{x: s(x)<+ {\infty} \}= \mathbb{R}\) modulo sets of Lebesgue zero, or its complement the divergence set \(D(f, {\Lambda})=\{x: s(x)=+ {\infty} \}= \mathbb{R}\) modulo sets of measure zero. If \({\Lambda}\) is not type 1 we say that \({\Lambda}\) is type 2.
The exact characterization of type 1 and type 2 sets is not known. In this paper we continue our study of the properties of type 1 and 2 sets. We discuss sub and supersets of type 1 and 2 sets and give a complete and simple characterization of a subclass of dyadic type 1 sets. We discuss the existence of type 1 sets containing infinitely many elements independent over the rationals. Finally, we consider unions and Minkowski sums of type 1 and 2 sets.
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Acknowledgements
Z. Buczolich thanks the Rényi Institute where he was a visiting researcher for the academic year 2017-18. B. Hanson would like to thank the Fulbright Commission, the Budapest Semesters in Mathematics, and the Rényi Institute for their generous support during the Spring of 2018, while he was visiting Budapest as a Fulbright scholar.
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Dedicated to the memory of Ákos Császár
Z. Buczolich was supported by the Hungarian National Research, Development and Innovation Office–NKFIH, Grant 124749.
B. Maga was supported by the ÚNKP-17-2 New National Excellence of the Hungarian Ministry of Human Capacities, and by the Hungarian National Research, Development and Innovation Office–NKFIH, Grant 124003.
G. Vértesy was supported by the Hungarian National Research, Development and Innovation Office–NKFIH, Grant 124749.
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Buczolich, Z., Hanson, B., Maga, B. et al. Type 1 and 2 sets for series of translates of functions. Acta Math. Hungar. 158, 271–293 (2019). https://doi.org/10.1007/s10474-019-00937-2
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DOI: https://doi.org/10.1007/s10474-019-00937-2