Large arithmetic progression modulo $p$

Question

Let $B$ be a subset of $Z_p(=\mathbb{Z}/p\mathbb{Z})$ of cardinal $Cp^{\frac{1}{3}}$, for some constant $C$. How to construct an arithmetic progression of length $C_1p^{\frac{2}{3}}$ where $C_1$ is some constant, inside $B+\alpha B$ for any $\alpha \in Z_p$ ?

Answer 1

The statement is not true.

Take $B$ to be a set of the form $$\{a + b M : 1 \leq a, b \leq N\},$$ where $(N+1)M <p$, $N^2 \approx p^{1/3}$ and $n < 2M$.

Then $B$ is Freiman isomorphic to the square $[1, N] \times [1,N] \cap \mathbb{Z}^2$ and so the longest arithmetic progression in $B+B$ is of size $\lesssim |B|^{1/2}$.

Take a look at this paper of Szemeredi and Vu.