AI 中文总结
本文突破Ruzsa方法的$3/4$指数壁垒，通过新构造得到无平方差集$D(N)$的下界指数达$0.7527964558\dots$，优于此前最优结果。
AI 中文摘要
令$D(N)$表示集合$\{1,\ldots,N\}$中不含非零平方差的子集的最大基数。虽然证明$D(N)\geq(1-o(1))N^{1/2}$的构造几乎是平凡的，但Erdős猜想该界在多对数因子范围内是紧的。这一猜想被Sárközy否定，之后Ruzsa又找到一种精巧的构造，证明$D(N)\geq c\cdot N^{0.733077\dots}$，其中$c>0$为绝对常数。他的方法后续得到改进，此前最优的下界指数为$0.7334117\dots$，由Beigel-Gasarch和Lewko分别独立得到。不过Ruzsa在原始论文中指出，$3/4$似乎是其方法的自然壁垒。本文提出一种新构造，得到下界：当$N\to\infty$时，$\frac{\log D(N)}{\log N}$的下极限不小于$\alpha_*:=0.7527964558\dots$，从而突破了Ruzsa方法的自然指数$3/4$壁垒。该值$0.7527964558\dots$源于一个简单的优化问题，且似乎是新方法的极限值。
英文摘要
Let $D(N)$ denote the largest cardinality of a subset of $\{1,\ldots,N\}$ containing no nonzero square difference. While a construction certifying $D(N)\geq (1-o(1))N^{1/2}$ is almost trivial, Erdős conjectured that this bound is sharp up to polylogarithmic factors. This was disproved by Sárközy and later again by Ruzsa, who found an elegant construction showing that $D(N)\geq c\cdot N^{0.733077\dots}$, with an absolute constant $c>0$. His approach was subsequently refined, leading to the previously best known lower bound with exponent $0.7334117\dots$ due to Beigel-Gasarch and, independently, Lewko. However, in the original paper Ruzsa observed that $3/4$ seems to be the natural barrier of his approach. In this paper we develop a new construction leading to the lower bound \[ \liminf_{N\to\infty}\frac{\log D(N)}{\log N} \geq α_*:= 0.7527964558\ldots; \] thus crossing the natural exponent-$3/4$ barrier of Ruzsa's method. The value $0.7527964558\ldots$ arises from a simple optimisation problem and appears to be the limit of the new approach.
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