理想光滑集合与实数积和指数的显式上界
Ideal-Smooth Sets and an Explicit Upper Bound for the Real Sum-Product Exponent
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中文总结 AI 辅助
本文构造任意大的实数代数整数集，使和集与积集均不超过$|A|^{1.95835}$，通过理想光滑$S$-单位与互素因子结合，并利用张量积秩和区间算术给出显式上界。
中文摘要 AI 辅助
我们构造了任意大的实数代数整数有限集合$A$，使得$|A+A|$和$|AA|$均至多为$|A|^{1.95835}$。该构造结合了截断的理想光滑$S$-单位纤维与一个互素的加法因子。利用张量积秩论证，在每个选定的素理想处采用二维局部特征，控制加法因子中的损失，而对外部平行体的直接估计改进了和集打包界。算术输入是已知十次域上的非分歧pro-$2$塔，同时通过Frobenius截断控制小素理想。我们使用无条件Tsfasman--Vlăduţ不等式来处理联合类数-正则子代价。指数中所有有限数值比较均由向外有理区间算术验证。未使用任何未经证明的假设。
英文摘要
We construct arbitrarily large finite sets $A$ of real algebraic integers such that both $|A+A|$ and $|AA|$ are at most $|A|^{1.95835}$. The construction combines truncated ideal-smooth $S$-unit fibres with a coprime additive factor. A tensor-product rank argument, using a two-dimensional local feature at each selected prime ideal, controls the loss in the additive factor, while a direct estimate for an outer parallel body improves the sumset packing bound. The arithmetic input is an unramified pro-$2$ tower over a known degree-ten field, with simultaneous Frobenius cuts controlling the small prime ideals. We use unconditional Tsfasman--Vlăduţ inequalities for the joint class-number--regulator cost. All finite numerical comparisons entering the exponent are certified by outward rational interval arithmetic. No unproved hypothesis is used.
