Exercise 24, § 1

(a) Let $\delta >0$. If $p$ is a sufficiently large prime number and if A is a subset of $\mathbf{Z}/p\mathbf{Z}$ such that Card(A) $\geqslant \delta p$, then A contains an arithmetic progression of length 3. (Adapt the method of the preceding exercise.) (b) Let $\delta >0$. If N is a sufficiently large integer and if $A\subset \{1, . . . ,N\}$ satisfies Card(A) $\geqslant \delta N$, then A contains an arithmetic progression of length 3. (Reduce to the case of the preceding question by choosing $p$ suitably as a function of N.)