  We give a solution of the problem on trigonometric polynomials $f_n$ with the
given leading harmonic $y\cos nt$ that deviate the least from zero in measure,
more precisely, with respect to the functional $\mu(f_n)=mes\{t\in[0,2\pi]:
|f_n(t)|\ge 1\}$. 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.
