Denote by $\Delta_M$ the $M$-dimensional simplex. A map $f\colon \Delta_M\to\mathbb R^d$ is an almost $r$-embedding if $f\sigma_1\cap\ldots\cap f\sigma_r=\emptyset$ whenever $\sigma_1,\ldots,\sigma_r$ are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if $r$ is not a prime power and $d\ge2r+1$, then there is an almost $r$-embedding $\Delta_{(d+1)(r-1)}\to\mathbb R^d$. This was improved by Blagojevi\'c-Frick-Ziegler using a simple construction of higher-dimensional counterexamples by taking $k$-fold join power of lower-dimensional ones. We improve this further (for $d$ large compared to $r$): If $r$ is not a prime power and $N:=(d+1)r-r\Big\lceil\dfrac{d+2}{r+1}\Big\rceil-2$, then there is an almost $r$-embedding $\Delta_N\to\mathbb R^d$. For the $r$-fold van Kampen-Flores conjecture we also produce counterexamples which are stronger than previously known. Our proof is based on generalizations of the Mabillard-Wagner theorem on construction of almost $r$-embeddings from equivariant maps, and of the \"Ozaydin theorem on existence of equivariant maps.

翻译：令$\Delta_M$表示$M$维单形。若对于任意两两不相交的面$\sigma_1,\ldots,\sigma_r$，总有$f\sigma_1\cap\ldots\cap f\sigma_r=\emptyset$，则称映射$f\colon \Delta_M\to\mathbb R^d$为几乎$r$-嵌入。拓扑Tverberg猜想的一个反例断言：若$r$不是素数幂且$d\ge2r+1$，则存在几乎$r$-嵌入$\Delta_{(d+1)(r-1)}\to\mathbb R^d$。Blagojević-Frick-Ziegler通过将低维反例进行$k$重联合幂的简单构造，改进了高维反例。我们进一步改进此结果（在$d$相对于$r$较大时）：若$r$不是素数幂且$N:=(d+1)r-r\Big\lceil\dfrac{d+2}{r+1}\Big\rceil-2$，则存在几乎$r$-嵌入$\Delta_N\to\mathbb R^d$。对于$r$重van Kampen-Flores猜想，我们也构造出比已知结果更强的反例。我们的证明基于Mabillard-Wagner定理（关于从等变映射构造几乎$r$-嵌入）与Özaydin定理（关于等变映射存在性）的推广。