On additive irreducibility of multiplicative subgroups

In this paper, we employ a version of Stepanov’s method, developed by Hanson and Petridis, to prove several results on additive irreducibility of multiplicative subgroups in finite fields of prime order $p$. Specifically, we show that if a subgroup ${\mu }_d$ of $d$-th roots of unity satisfies $A-A={\mu }_d\cup \{0\}$, then $d=2$ or $6$. Additionally, we resolve the Sárközy’s conjecture on quadratic residues: for prime $p$, the set ${\mathcal{R}}_p$ of quadratic residues modulo $p$ cannot be represented as $A+B$ for $A,B$ with $\min (\mathrm{\mid }A\mathrm{\mid },\mathrm{\mid }B\mathrm{\mid })>1$. More generally, we prove that if the set of $d$-th roots of unity ${\mu }_d$ is represented non-trivially as $A+B$, then the sizes of summands are equal.