Remark on the theory of Sergeev frequencies of zeros, signs, and roots for solutions of linear differential equations: II

Abstract

The theorem that claims that the spectra (ranges) of upper and lower Sergeev frequencies of zeros, signs, and roots of a linear differential equation of order > 2 with continuous coefficients belong to the class of Suslin sets on the nonnegative half-line of the extended numerical line is inverted for the spectra of upper frequencies of third-order equations under the assumption that the spectra contain zero. In addition, we construct examples of third-order equations with continuous coefficients whose Lebesgue sets of the upper Sergeev frequency of signs belong to the exact first Borel class, and the Lebesgue sets of upper Sergeev frequencies of zeros and roots belong to the exact second Borel class.