ALL-SUMS SETS IN (0, 1] - CATEGORY AND MEASURE

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. Tn fact any set which is large around 0 in the sense of measure or category will contain such a sequ ence. We show that sets with 0 as a density point have very rich struc ture. Call a sequence [t(n)](n=1)(infinity) and its resulting all-sums set structured provided for each n, t(n) greater than or equal to Sig ma(k=n+1)(infinity) t(k). We show further that structured all-sums set s with positive measure are not partition regular even if one allows s hifted all-sums sets. That is, we produce a two cell measurable partit ion of (0, 1] such that neither set contains a translate of any struct ured all-sums set with positive measure.