Question 41
Let $n$ be a positive integer and $A$ a subset of $\mathbb{Z}/(n)$, the set of the integers modulo $n$, define $f(A) = \min_{t \in \mathbb{Z}/(n)} |A \cap (A+t)|$, where $A+t = \{x+t, x \in A\} \subset \mathbb{Z}/(n)$. Define $g(n) = \max\{f(A); A \subset \mathbb{Z}/(n), |A| = \lfloor n/2 \rfloor\}$.
a. Prove that $g(n) \le \lfloor n/4 \rfloor - 1, \forall n \ge 1$.
b. Prove that $g(n) = \lfloor n/4 \rfloor - 1$ for infinite values of $n \ge 1$.
a. Prove that $g(n) \le \lfloor n/4 \rfloor - 1, \forall n \ge 1$.
b. Prove that $g(n) = \lfloor n/4 \rfloor - 1$ for infinite values of $n \ge 1$.