 Bose-Chowla Sidon construction

Choose a generator $\theta$ of the multiplicative group of the quadratic finite field extension. Its $p$ shifts $\theta +t$ define $p$ distinct exponents modulo $p^2-1$. Equality of two sums of exponents gives $(\theta +s)(\theta +t)=(\theta +u)(\theta +v)$. Since $1,\theta$ are independent over the prime field, $s+t=u+v$ and $st=uv$, so the unordered pairs coincide. Thus the exponents form a Sidon set in the cyclic group. Taking integer representatives preserves the Sidon property. Primes within $o(\sqrt{N})$ below $\sqrt{N}$ give $\sqrt{N}-o(\sqrt{N})$ representatives in an interval of length $N$.