On the distribution of positive and negative values of Hardy's $Z$-function

We investigate the distribution of positive and negative values of Hardy’s function $$Z(t):=\zeta (1\mathrm{/}2+it){\chi (1\mathrm{/}2+it)}^{-1\mathrm{/}2},\zeta (s)=\chi (s)\zeta (1-s)\mathrm{.}$$ In particular we prove that $$\mu (I_{+}(T,T))\gg T\text{and}\mu (I_{-}(T,T))\gg T,$$ where $\mu (\cdot )$ denotes the Lebesgue measure and \begin{align*} { I}+(T,H) &;=; \bigl{T< t\le T+H,:, Z(t)>0\bigr}, { I}-(T,H) &;=; \bigl{T< t\le T+H,:, Z(t)<0\bigr}. \end{align*}