Recently, several studies have shown that when $q\equiv3\pmod{4}$, the function $F_r(x)=x^r+x^{r+\frac{q-1}{2}}$ defined over $\mathbb{F}_q$ 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 $x\in \mathbb{F}_q$ with $χ(x)=χ(x+1)=1$ satisfying $(x+1)^r - x^r = b$ for all $b\in \mathbb{F}_q^*$ 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：最近的研究表明，当$q\equiv3\pmod{4}$时，定义在$\mathbb{F}_q$上的函数$F_r(x)=x^r+x^{r+\frac{q-1}{2}}$具有局部APN性质，且其回旋一致性至多为~$2$。本文推广了这些结果，证明若对所有$b\in \mathbb{F}_q^*$，满足$(x+1)^r - x^r = b$且$\chi(x)=\chi(x+1)=1$的$x\in \mathbb{F}_q$至多有一个，且$\gcd(r,q-1)\mid 2$，则$F_r$为局部APN函数且其回旋一致性至多为$2$。此外，我们研究了$F_3$与$F_{\frac{2q-1}{3}}$的差分谱，以及$p=3$时$F_2$的回旋谱。