In their paper Sign changes of Hecke eigenvalues, Matomaki and Radziwill showed that (Theorem 1.2 of the paper) for a large enough $x$ , the number of sign changes of sign changes of in the non-vanishing sequence of Hecke eigenvalues $(\lambda_f(n))_{n\leq x}$ is $\asymp \frac{x}{k(x)},$ where $$k(x)=\prod_{\substack{p \leq x\\ \lambda_f(p)=0}} \left(1+\frac{1}{p}\right).$$ In the proof of this result, they used the fact that there is a positive proportion of $x$ in $[X,2X]$ such that $$\left|\sum_{x\leq n \leq x+h k(x)} \operatorname{sgn}(\lambda_f(n))w_{n}\right|< \sum_{x\leq n \leq x+h k(x)} w'_{n}$$ once $h$ is large enough but satisfying some condition. They say that since $w'_n \leq w_n$ for every $n$ then for every $x$ satisfying the upper bound then a sign change of $\lambda_f(n)$ must occur in the interval $[x, x+h k(x)]$. From which they say that there are $\gg \frac{X}{h(X)}$ sign changes.

I have two questions:

1. I could not understand how they get their deduction about the lower bound of sign changes.

2. Why they used the same proof to deduce an upper bound although they did not mention it!

Did I misunderstand something?

In their paper Sign changes of Hecke eigenvalues, Matomaki and Radziwill showed that (Theorem 1.2 of the paper) for a large enough $x$ , the number of sign changes of sign changes of in the non-vanishing sequence of Hecke eigenvalues $(\lambda_f(n))_{n\leq x}$ is $\asymp \frac{x}{k(x)},$ where $$k(x)=\prod_{\substack{p \leq x\\ \lambda_f(p)=0}} \left(1+\frac{1}{p}\right).$$ In the proof of this result, they used the fact that there is a positive proportion of $x$ in $[X,2X]$ such that $$\left|\sum_{x\leq n \leq x+h k(x)} \operatorname{sgn}(\lambda_f(n))w_{n}\right|< \sum_{x\leq n \leq x+h k(x)} w'_{n}$$ once $h$ is large enough but satisfying some condition. They say that since $w'_n \leq w_n$ for every $n$ then for every $x$ satisfying the upper bound then a sign change of $\lambda_f(n)$ must occur in the interval $[x, x+h k(x)]$. From which they say that there are $\gg \frac{X}{k(X)}$ sign changes.

I have two questions:

1. I could not understand how they get their deduction about the lower bound of sign changes.

2. Why they used the same proof to deduce an upper bound although they did not mention it!

Did I misunderstand something?

In their paper Sign changes of Hecke eigenvalues, Matomaki and Radziwill showed that (Theorem 2.1 of the paper) for a large enough $x,$ the number of sign changes of sign changes of in the non-vanishing sequence of Hecke eigenvalues $(\lambda_f(n))_{n\leq x}$ is $\asymp \frac{x}{k(x)},$ where $$k(x)=\prod_{\substack{p \leq x\\ \lambda_f(p)=0}} \left(1+\frac{1}{p}\right).$$ In the proof of this result, they used the fact that there is a positive proportion of $x$ in $[X,2X]$ such that $$\left|\sum_{x\leq n \leq x+h k(x)} sign(\lambda_f(n))w_{n}\right|< \sum_{x\leq n \leq x+h k(x)} w'_{n} $$ once $h$ is large enough but satisfying some condition. They say that since $w'_n \leq w_n$ for every $n$ then for every $x$ satisfying the upper bound then a sign change of $\lambda_f(n)$ must occur in the interval $[x, x+h k(x)].$ Form which they say that there are $\gg \frac{X}{h(X)}$ sign changes.

In fact, I have two questions:

First, I could not understand how they get their deduction about the lower bound of sign changes.

Second, why they used the same proof to deduce an upper bound although they did not mention it! Did I misunderstand something? Thanks in advance.

In their paper Sign changes of Hecke eigenvalues, Matomaki and Radziwill showed that (Theorem 1.2 of the paper) for a large enough $x$ , the number of sign changes of sign changes of in the non-vanishing sequence of Hecke eigenvalues $(\lambda_f(n))_{n\leq x}$ is $\asymp \frac{x}{k(x)},$ where $$k(x)=\prod_{\substack{p \leq x\\ \lambda_f(p)=0}} \left(1+\frac{1}{p}\right).$$ In the proof of this result, they used the fact that there is a positive proportion of $x$ in $[X,2X]$ such that $$\left|\sum_{x\leq n \leq x+h k(x)} \operatorname{sgn}(\lambda_f(n))w_{n}\right|< \sum_{x\leq n \leq x+h k(x)} w'_{n}$$ once $h$ is large enough but satisfying some condition. They say that since $w'_n \leq w_n$ for every $n$ then for every $x$ satisfying the upper bound then a sign change of $\lambda_f(n)$ must occur in the interval $[x, x+h k(x)]$. From which they say that there are $\gg \frac{X}{h(X)}$ sign changes.

I have two questions:

1. I could not understand how they get their deduction about the lower bound of sign changes.

2. Why they used the same proof to deduce an upper bound although they did not mention it!

Did I misunderstand something?