Let $q$ be an odd prime power with $q\equiv 3\ ({\rm{mod}}\ 4)$. In this paper, we study the differential and boomerang properties of the function $F_{2,u}(x)=x^2\big(1+u\eta(x)\big)$ over $\mathbb{F}_{q}$, where $u\in\mathbb{F}_{q}^*$ and $\eta$ is the quadratic character of $\mathbb{F}_{q}$. We determine the differential uniformity of $F_{2,u}$ for any $u\in\mathbb{F}_{q}^*$ and determine the differential spectra and boomerang uniformity of the locally-APN functions $F_{2,\pm 1}$, thereby disproving a conjecture proposed in \cite{budaghyan2024arithmetization} which states that there exist infinitely many $q$ and $u$ such that $F_{2,u}$ is an APN function.

翻译：令 $q$ 为满足 $q\equiv 3\ ({\rm{mod}}\ 4)$ 的奇素数幂。本文研究定义在 $\mathbb{F}_{q}$ 上的函数 $F_{2,u}(x)=x^2\big(1+u\eta(x)\big)$ 的差分性质与回旋镖性质，其中 $u\in\mathbb{F}_{q}^*$，$\eta$ 为 $\mathbb{F}_{q}$ 上的二次特征。我们确定了任意 $u\in\mathbb{F}_{q}^*$ 下 $F_{2,u}$ 的差分均匀度，并确定了局部 APN 函数 $F_{2,\pm 1}$ 的差分谱与回旋镖均匀度，从而否定了 \cite{budaghyan2024arithmetization} 中提出的猜想——该猜想声称存在无穷多组 $q$ 与 $u$ 使得 $F_{2,u}$ 为 APN 函数。