Variations on a theme of Debs and Saint Raymond

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