Recently, several studies have shown that when $q\equiv3\pmod{4}$, for certain choices of $r$, the function $F_r(x)=x^r+x^{r+\frac{q-1}{2}}$ defined over $\Fq$ is locally-APN and has boomerang uniformity at most~$2$. In this paper, we extend these results by showing that if there is at most one $x\in \Fq$ with $χ(x)=χ(x+1)=1$ satisfying $(x+1)^r - x^r = b$ for all $b\in \Fqmul$ and $\gcd(r,q-1)\mid 2$, then $F_r$ is locally-APN with boomerang uniformity at most $2$. Moreover, we study the differential spectra of $F_3$ and $F_{\frac{2q-1}{3}}$, and the boomerang spectrum of $F_2$ when $p=3$.

翻译：近年来，多项研究表明，当$q\equiv3\pmod{4}$时，对于特定选择参数$r$，定义在$\Fq$上的函数$F_r(x)=x^r+x^{r+\frac{q-1}{2}}$是局部APN的，且其回旋均匀性至多为$2$。本文通过证明以下结论推广了这些结果：若至多存在一个$x\in \Fq$满足$\chi(x)=\chi(x+1)=1$且使得对所有$b\in \Fqmul$均有$(x+1)^r - x^r = b$成立，且$\gcd(r,q-1)\mid 2$，则$F_r$是局部APN的且回旋均匀性至多为$2$。此外，我们还研究了当$p=3$时$F_3$和$F_{\frac{2q-1}{3}}$的差分谱，以及$F_2$的回旋谱。