All-sums sets in 0, 1] - Category and measure

Vitaly Bergelson*, Neil Hindman, Benjamin Weiss

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

We provide a unified and simplified proof that for any partition of (0, 1] into sets that are measurable or have the property of Baire, one cell will contain an infinite sequence together with all of its sums (finite or infinite) without repetition. In fact any set which is large around 0 in the sense of measure or category will contain such a sequence. We show that sets with 0 as a density point have very rich structure. Call a sequence 〈tnn=1 and its resulting all-sums set structured provided for each n, tn≥ Σk=n+1 tk. We show further that structured all-sums sets with positive measure are not partition regular even if one allows shifted all-sums sets. That is, we produce a two cell measurable partition of (0, 1] such that neither set contains a translate of any structured all-sums set with positive measure.

Original languageEnglish
Pages (from-to)61-87
Number of pages27
JournalMathematika
Volume44
Issue number1
DOIs
StatePublished - Jun 1997

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