Sets of Divergence for Fourier Series

There exists a G set such that no Fourier series can divergeon it and converge off it.     

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.     

Selecting non-consecutive balls arranged in many lines

Kaplansky obtained the number of ways of selecting k balls from n balls arranged in a line without two selected ones being consecutive. Here his result is generalized to the case of any number of lines where each line can contain any number of balls. Several interesting corollaries are derived; one of them reduces to a result of Konvalina's.     

Differentiable Functions on the Rationals

We construct a smooth bijective function from the rationalsto the rationals whose inverse is nowhere continuous.     

A Theorem of Mensov on the Adjustment of Functions

A new proof is given of a theorem of Menov which states thata measurable function can be changed on a set of arbitrarilysmall measure to obtain a function with uniformly convergentFourier series. Other results of Menov on the representationof functions by trigonometric series are obtained by using arelated result.     

A Topological Ivasev-Musatov Theorem

If the sum of a series of positive numbers is not convergentand the series satisfies certain regularity conditions, a probabilitymeasure can be found whose support contains no interval, yetwhose Fourier series decreases faster than the given series.