Trigonometric polynomials of least deviation from zero in measure and related problems

We give a solution of the problem on trigonometric polynomials fn with the given leading harmonic ycosnt that deviate the least from zero in measure, more precisely, with respect to the functional μ(fn)=mes{t∈[0,2π]:|fn(t)|≥1}. For trigonometric polynomials with a fixed leading harmonic, we consider the least uniform deviation from zero on a compact set and find the minimal value of the deviation over compact subsets of the torus that have a given measure. We give a solution of a similar problem on the unit circle for algebraic polynomials with zeros on the circle.