On the convergence in category of trigonometric series (Q1329603)

The main result of the paper is the following theorem related to the classical theorem of Menchoff: If \(f\) is a continuous function on \([- \pi,\pi]\), then for each \(\varepsilon> 0\) there exists a set \(X\) of second category such that \(m([-\pi,\pi]- X)<\varepsilon\) and a trigonometric series which converges to \(f\) on \(X\) has uniformly bounded partial sums. The result cannot be improved to obtain a set of full measure.