We construct $n$-node graphs on which any $O(n)$-size spanner has additive error at least $+\Omega(n^{3/17})$, improving on the previous best lower bound of $\Omega(n^{1/7})$ [Bodwin-Hoppenworth FOCS '22]. Our construction completes the first two steps of a particular three-step research program, introduced in prior work and overviewed here, aimed at producing tight bounds for the problem by aligning aspects of the upper and lower bound constructions. More specifically, we develop techniques that enable the use of inner graphs in the lower bound framework whose technical properties are provably tight with the corresponding assumptions made in the upper bounds. As an additional application of our techniques, we improve the corresponding lower bound for $O(n)$-size additive emulators to $+\Omega(n^{1/14})$.

翻译：我们构造了$n$节点图，使得任意$O(n)$大小的斯潘纳的加法误差至少为$+\Omega(n^{3/17})$，改进了先前最佳下界$\Omega(n^{1/7})$ [Bodwin-Hoppenworth FOCS '22]。我们的构造完成了特定三步研究计划的前两步，该计划在先前工作中引入并在此概述，旨在通过对齐上界和下界构造的各个方面来产生问题的紧界。更具体地说，我们开发了技术，使得在下界框架中使用内图成为可能，这些内图的技术性质被证明与上界中相应的假设是紧的。作为我们技术的额外应用，我们将$O(n)$大小加法仿真器的相应下界改进为$+\Omega(n^{1/14})$。