We study the sums of nondegenerate sine series with monotone coefficients and consider the sets of positivity of such functions. We obtain the sharp lower estimate of the measure of such a set on $$[\pi/2, \pi]$$ [ π / 2 , π ] and a new lower bound on its measure on $$[0,\pi]$$ [ 0 , π ] . It is shown that the latter measure is at least $$\pi/2 + 0.24$$ π / 2 + 0.24 and in the case of fulfilling special conditions it is at least $$2\pi/3$$ 2 π / 3 , which is an unimprovable estimate.

中文翻译：

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带单调系数的正弦级数的正集

我们研究具有单调系数的非退化正弦级数的和，并考虑这些函数的正集。我们在 $$[\pi/2, \pi]$$ [ π / 2 , π ] 上获得了这样一个集合的测度的急剧下限估计以及它在 $$[0,\pi 上的测度的新下界]$$ [ 0 , π ] 。结果表明，后一种度量至少为 $$\pi/2 + 0.24$$ π / 2 + 0.24 并且在满足特殊条件的情况下至少为 $$2\pi/3$$ 2 π / 3 ，这是一个不可改进的估计。