Unique-neighbor expander

Question

I want to solve Problem 4.10 from Randomness by Salil Vadhan. https://people.seas.harvard.edu/~salil/cs225/spring15/PS3.pdf

Consider a bipartite expander $G$ with left degree $D$ so that every subset $S$ of the left vertices with at most $K$ vertices has at least $(1-\epsilon)D|S|$ neighbors. Then $G$ also has the property that it has $(1-2\epsilon)D|S|$ unique neighbors. Unique meaning that it has exactly one corresponding vertex from $S$.

I recognize that the new expansion factor is $(1-2\epsilon)D = 2\cdot(1-\epsilon)D -D$

Answer 1

Let $S$ be a subset of at most $K$ vertices. Let $a_d$ be the number of vertices on the right side which are connected to exactly $d$ vertices in $S$. Since the left degree is $D$, $$ \sum_{d \geq 1} da_d = D|S|. $$ Since $S$ has at least $(1-\epsilon)D|S|$ neighbors, $$ \sum_{d \geq 1} a_d \geq (1-\epsilon)D|S|. $$ Therefore $$ \sum_{d \geq 2} a_d \leq \sum_{d \geq 1} (d-1) a_d = \sum_{d \geq 1} da_d - \sum_{d \geq 1} a_d \leq D|S| - (1-\epsilon)D|S| = \epsilon D|S|. $$ It follows that $$ a_1 = \sum_{d \geq 1} a_d - \sum_{d \geq 2} a_d \geq (1-\epsilon)D|S| - \epsilon D|S| = (1-2\epsilon)D|S|. $$