Playing with the Szemerédi's theorem

The Szemeredi's theorem states that, for every strictly positive integer $k$ and real number $0 < \delta \le \frac 12$ , there exists a positive integer $N = N(k, \delta)$ such that every subset of $\{1, 2, ..., N\}$ of size at least $N \times \delta$ contains an arithmetic progression of length $k$ .

For a given $k$ , a given $\delta$ and $N$ the smallest possible number that satisfies the statement above, is $N \times \delta$ an integer?