I am confused about the covering lemmas in the captioned work and really hope to get some ideas here.
Firstly it is lemma 3.7. (Image of Lemma 3.7) (for convenience here is the lemma of this lemma (Image of Lemma 3.6))
I do not understand the ''$\supseteq$'' in the line ''$J'\cap(J'-\ell)\supseteq \cdots$''. Also I am wondering if the following provides a counter-example of this line and this lemma:
Let $n=12$, $I=\{0,1,2,3,4,5,6,7,8\}$, $m=3$. Then $I'=\{0,4,8\}$ by the construction in Lemma 3.6. Let $J=\{0,2,4,6,8,10,12\}$, $\delta=0.5$. Then for all $i\in I$, $|[i,i+3]\cap J|=2\geq(1-\delta)m$. Take $\ell=1$. Then for all $J'\subseteq J$, we have $J'\cap(J'-\ell)\subseteq J\cap(J-\ell)=\emptyset$, $\bigcup_{i\in I'} J\cap[i,i+2]=\{0,2,4,6,8,10\}$, and $(1-\delta-\frac{\ell}{m})|I|=(\frac{1}{2}-\frac{1}{3})\cdot 9 >0$.
Secondly, it is lemma 3.8.(Excerpt for Lemma 3.8), which I am not sure how the marked inequality is obtained.
Since subsequent results depends on these lemmas and this paper is already acknowledged in the field, I think there should be an answer for resolving my confusion.
Thanks for help!
