$r$.

\(iii) For all $i\in {1,2,\cdots,l}$, we have $$\sum_{r=0}^{K}\#(Sp\phi_{x_0}\cap V_r^i)\geq\frac{L}{L+1}\#(Sp\phi_{x_0}\cap (0,1)_i).$$

Set $t_0'=0,\ t_1'=\frac{1}{K},\cdots,\,t_K'=\frac{K}{K}=1$. Consider each set $(t_{r-1}',t_r')_i$, we will define intermediate points $S_r,\widetilde{S}_r$ with $t_{r-1}'\leq S_r \leq \widetilde{S}_r\leq t_r'$, $(1\leq r\leq K)$ as follows.

Divide $(t_{r-1}',t_r')_i$ into $L+1$ intervals of equal length $\frac{1}{K(L+1)}$ by points $$t_{r-1}'=\gamma_0<\gamma_1<\cdots<\gamma_{L+1}=t_r'.$$ Comparing $\#\big(Sp\phi_{x_0}\cap (\gamma_k,\gamma_{k+1})_i\big)$ for $k=0,1,\cdots,L$, choose $0\leq k_0\leq L$ such that $$\#\big(Sp\phi_{x_0}\cap(\gamma_{k_0},\gamma_{k_0+1})_i\big)\leq\#\big(Sp\phi_{x_0}\cap(\gamma_k,\gamma_{k+1})_i\big)$$ for each $0\leq k\leq R-1$. Set $S_r=\gamma_{k_0}$ and $\widetilde{S_r}=\gamma_{k_0+1}$. Then $$\#\big(Sp\phi_{x_0}\cap(S_r,\widetilde{S_r})_i\big)\leq\frac{1}{L+1}\#\big(Sp\phi_{x_0}\cap(t_{r-1},t_r)_i\big).$$ Hence $$\sum_{r=1}^{K}\#\big(Sp\phi_{x_0}\cap(S_r,\widetilde{S_r})_i\big)\leq \frac{1}{L+1}\#(Sp\phi_{x_0}\cap(0,1)_i)=\frac{1}{L+1}\#(Sp\phi_{x_0}\cap (0,1)_i).$$ Set $\widetilde{V}=\cup_{j=1}^K(S_j,\widetilde{S_j})$, then $$\#\big (Sp\phi_{x_0}\cap [0,1]_i \backslash \widetilde{V}\big)\geq \frac{L}{L+1}\#(Sp\phi_{x_0}\cap (0,1)_i).$$ Since $[0,1]_i \backslash \widetilde{V}$ is a disjoint union of closed intervals $V_0^i,V_1^i,V_2^i,\cdots,V_K^i$, we note that some of them may be single point. Actually, we denote them of the form $$V_0^i=[0,S_1]_i,V_1^i=[\widetilde{S_1},S_2]_i,\cdots,V_K^i=[\widetilde{S_K},1]_i.$$

Condition (iii) has already be verified. Let us verify Conditions (i) and (ii).

Since $\widetilde{S_r}-S_r=\dfrac{1}{K(L+1)}$, $$dist(V_{r_1}^i,V_{r_2}^i)\geq\frac{1}{K(L+1)},\quad if\quad r_1\neq r_2.$$ From the definition of $W_r^i$, we know that $$dist(W_{r_1}^i,W_{r_2}^i)\geq distance(V_{r_1}^i,V_{r_2}^i)-\frac{2}{4K(L+1)}\geq\frac{1}{2K(L+1)}.$$ Hence $W_{r_1}^i\cap W_{r_2}^i=\varnothing$ if $r_1\neq r_2$. This means (i) holds.

For (ii), notice that for each $r$ $$t_{r-1}'+\frac{1}{K(L+1)}\leq \widetilde{S_r}\leq S_{r+1}\leq  t_{r+1}'-\frac{1}{K(L+1)}.$$ Hence, $$diameter (V_r^i)\leq\frac{2}{K}- \frac{2}{K(L+1)}$$ It follows that $$diameter(W_r^i)<\frac{2}{K}.$$ So (ii) holds.

For each $i\in \{1,2,\cdots,l\}$, we have a collection of disjoint closed intervals $$V_0^1,V_1^1,\cdots,V_{K}^1,\cdots,V_0^l,V_1^l,\cdots,V_K^l$$ that satisfies conditions (i), (ii), (iii), and $V_r^i\subset [\eta,1-\eta]_i$ for all $r=\{1,2,\cdots,K-1\}$.

