Additive decompositions of cubes in finite fields

Let $$p\equiv 1\pmod 3$$ be a prime . We study several topics on additive decompositions concerning the set $$C_p$$ of all non-zero cubes in the finite field of p elements. For example, we show that when $$p>184291$$ , the set $$C_p$$ has no decomposition of the form $$C_p=A+B+C$$ with $$|A|,|B|,|C|\ge 2$$ .