Variations on a theme of Debs and Saint Raymond

A famous theorem of Debs and Saint Raymond states that the complementof a set of first category is of strong multiplicity. We provea theorem which combines this with a result of Rudin which statesthat independent closed sets of strong multiplicity exist. Wealso prove a theorem which combines the theorem of Debs andSaint Raymond with a theorem of Wiener and Wintner which statesthat there exists a measure with singular support whose convolutionsquare is absolutely continuous.