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In their paper 'Sign changes of Hecke eigenvalues', Matomaki and Radziwill established in Lemma 6.2 the following result: There exists absolute positive constants $c$ and $\eta$ such that uniformly in $h \leq X^{\eta},$ we have $$\int_{X}^{2X} \left|\sum_{x\leq n\leq x+hk(X)} sign(\lambda_f(n)) w_n \right|^2 dx \leq c^2h X,$$ where $k(x)=\displaystyle{\prod_{\substack{p \leq x\\ \lambda_f(p)=0}}}\left(1+\frac{1}{p}\right)$ and $w_n$ is the sieve weight then they used this Lemma to prove Proposition 3.4. In the proof of this proposition, they used Chebyshev's inequality to prove that the proportion of sign changes is at least $(1-1/K^2)$ where $K$ is as in the proof of Proposition 3.4 is such that the measure of the set of $x$ in $[X,2X]$ for which
$$\left|\sum_{x\leq n\leq x+hk(x)} sign(\lambda_f(n) w_n \right|\geq cK \sqrt{h} $$ is by Chebyshev's inequality $\leq X/K^2.$ Can someone clarify to me this proof I mean give me the relation or the expression of $K$ in terms of $c$? Thanks in advance.

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    $\begingroup$ In future, please use a more descriptive title for your questions. $\endgroup$ – Joe Silverman May 24 '16 at 11:47
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The constant $c>0$ in Lemma 6.2 is absolute, while $K>0$ is arbitrary in Proposition 3.4. There is no relation between these two quantities.

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