AI 中文总结
研究关于无对称着色 \(k\) 项等差数列的着色问题，通过结合进位控制着色与分层域范数映射给出一种着色方法，改进了 4 - 等差数列着色上界指数，反驳了高尔斯对于偶数 \(k\geq6\) 的猜想下界，还在相关问题上取得进展。
AI 中文摘要
给定一种着色 \(c\) 和一个偶数 \(k\geq4\)，若一个非平凡的 \(k\) 项等差数列 \(a,a + d,\ldots,a+(k - 1)d\) 满足 \(c(a+(i - 1)d)=c(a+(k - i)d)\) 对所有 \(i\in[k/2]\) 成立，则称其为对称着色。邓、蒂多尔和赵提出 \([N]\) 是否能有 \(N^{o(1)}\) 种颜色的着色且无此类 4 - 等差数列，并给出了 \([N]\) 的一种 \(O(N^{\log_{22}3})\) 着色。我们给出了对于每个偶数 \(k\geq4\) 和每个大于 \(k\) 的素数 \(p\)，\(\mathbb{Z}/p^{k^2/4}\mathbb{Z}\) 的一种 \(O_k(p)\) 着色且无此类 \(k\) - 等差数列，从而得到 \([N]\) 的一种 \(O(N^{4/k^2})\) 着色，将 4 - 等差数列上界的指数从 \(\log_{22}3\) 改进到 \(1/4\)。该构造结合了 \(p\) 进制数字的进位控制着色和分层域范数映射。对于 4 - 等差数列的结果与贝伦德式乘积着色一起，在厄尔多斯关于用至少三种颜色给每个非平凡 4 - 等差数列着色的问题 160 中给出 \(h(N)\leq N^{1/4 + o(1)}\)。此结果还给出对于每个 \(\varepsilon>0\)，\(\rho_4(\alpha)=O_\varepsilon(\alpha^{5 - \varepsilon})\)，改进了关于鲁扎问题的界。我们对于 \(k\) - 等差数列的结果首次反驳了高尔斯对于所有偶数 \(k\geq6\) 的猜想下界。
英文摘要
Given a coloring $c$ and an even $k\ge 4$, a nontrivial $k$-term arithmetic progression~($k$-AP) $a,a+d,\ldots,a+(k-1)d$ is called symmetrically colored if $c(a+(i-1)d)=c(a+(k-i)d)$, $\forall i\in[k/2]$. Deng, Tidor, and Zhao asked whether $[N]$ admits a coloring with $N^{o(1)}$ colors and no such 4-APs, and gave an $O(N^{\log_{22}3})$-coloring of $[N]$. We give an $O_k(p)$-coloring of $\mathbb Z/p^{k^2/4}\mathbb Z$ without such $k$-APs for every even $k\ge 4$ and every prime $p>k$, and hence an $O_k(N^{4/k^2})$-coloring of $[N]$, improving the exponent in the upper bound for $4$-APs from $\log_{22}3$ to $1/4$. The construction combines a carry-control coloring of base-$p$ digits with a layered field norm mapping. Together with Behrend-style product colorings, our result for $4$-APs gives $h(N)\leq N^{1/4+o(1)}$ in Erdős's Problem~160 on coloring every nontrivial 4-AP with at least three colors. This result also yields $ρ_4(α)=O_\varepsilon(α^{5-\varepsilon})$ for every $\varepsilon>0$, improving the bound toward Ruzsa's question. Our result for $k$-APs disproves Gowers' conjectured lower bound for all even $k\ge6$ for the first time.
Comments7 pages, comments are welcome. Revised version adds several references
