发表机构
Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对多项式源与特定目标分布的指数级小重叠问题，通过放大非二进参数接受概率的均匀分离，构造了显式目标分布并证明了对任意固定次数的多项式源重叠上界，同时给出了相邻次数间的确定性层级关系。
AI 中文摘要
一个次数为$d$的多项式源是次数至多为$d$的多项式映射在$\mathbb{F}_2$上作用于任意多个均匀随机比特的输出。Khodabandeh和Shinkar（FOCS '26）证明了$\mathrm{Ber}(1/3)^{\otimes N}$与每个常数次多项式源的统计距离为$1-o(1)$，并猜想其重叠为指数级小。独立于Khodabandeh和Shinkar，Byramji、Kane、Morris和Ostuni（RANDOM '26）提出了一个距离为$1-\exp(-N^{\Omega_d(1)})$的显式目标分布的问题。我们解决了这两个问题。对于每个固定的$d\geq1$，每个次数为$d$的多项式源与$\mathrm{Ber}(1/3)^{\otimes N}$的重叠至多为$\exp(-c_dN)$，其中$c_d>0$与种子长度无关。对于二次多项式，$c_2=2^{-26}$就足够了。我们放大了Khodabandeh和Shinkar关于非二进参数（即不能表示为$a/2^b$（其中$a$和$b\geq0$为整数）的数）的接受概率的均匀分离。该结果推广到其他非二进伯努利参数以及作为有界多个有界次数多项式的布尔函数的坐标。我们还给出了相邻次数之间的均匀确定性层级关系。将$d+1$个输入上的不相交AND门的输出附加到均匀种子比特上，得到平坦的次数为$(d+1)$的目标分布，其熵为$k$，与每个次数为$d$的源的重叠为$\exp(-\Omega_d(\min\{k,N-k\}))$，其中$\min\{k,N-k\}\geq2(d+1)$。对于固定的$d$，在平坦目标分布中，这种熵依赖在指数上是最优的（至多差常数倍）。该构造具有局部性$d+1$，并使用$O(N)$次域运算来采样。当$k=\lfloor N/2\rfloor$时，对于固定的$0<\varepsilon<1$，它处理$d\leq(1-\varepsilon)\log_2N/3$且重叠为$\exp(-N^{\varepsilon-o(1)})$。证明结合了Gowers一致性范数的单调性、随机仿射立方体中点的两两独立性以及相对熵。
英文摘要
A degree-$d$ polynomial source is the output of a polynomial map of degree at most $d$ over $\mathbb{F}_2$ on arbitrarily many uniform random bits. Khodabandeh and Shinkar (FOCS '26) proved that $\mathrm{Ber}(1/3)^{\otimes N}$ has statistical distance $1-o(1)$ from every constant-degree polynomial source and conjectured exponentially small overlap. Independently of Khodabandeh and Shinkar, Byramji, Kane, Morris, and Ostuni (RANDOM '26) asked for an explicit target distribution at distance $1-\exp(-N^{Ω_d(1)})$. We resolve both questions. For every fixed $d\geq1$, every degree-$d$ polynomial source has overlap at most $\exp(-c_dN)$ with $\mathrm{Ber}(1/3)^{\otimes N}$, where $c_d>0$ is independent of the seed length. For quadratics, $c_2=2^{-26}$ suffices. We amplify Khodabandeh and Shinkar's uniform separation of acceptance probabilities from non-dyadic parameters (numbers not of the form $a/2^b$ for integers $a$ and $b\geq0$). The result extends to other non-dyadic Bernoulli parameters and to coordinates that are Boolean functions of boundedly many bounded-degree polynomials. We also give a uniform deterministic hierarchy between adjacent degrees. Appending the outputs of disjoint AND gates on $d+1$ inputs to uniform seed bits yields flat degree-$(d+1)$ target distributions of entropy $k$ with overlap $\exp(-Ω_d(\min\{k,N-k\}))$ against every degree-$d$ source, for $\min\{k,N-k\}\geq2(d+1)$. This entropy dependence is optimal up to constants in the exponent among flat target distributions for fixed $d$. The construction has locality $d+1$ and uses $O(N)$ field operations to sample. At $k=\lfloor N/2\rfloor$, it handles $d\leq(1-\varepsilon)\log_2N/3$ with overlap $\exp(-N^{\varepsilon-o(1)})$ for fixed $0<\varepsilon<1$. The proof combines monotonicity of Gowers uniformity norms, pairwise independence of points in a random affine cube, and relative entropy.
Comments17 pages
