Recently, two extraordinary results on aperiodic monotiles have been obtained in two different settings. One is a family of aperiodic monotiles in the plane discovered by Smith, Myers, Kaplan and Goodman-Strauss in 2023, where rotation is allowed, breaking the 50-year-old record (aperiodic sets of two tiles found by Roger Penrose in the 1970s) on the minimum size of aperiodic sets in the plane. The other is the existence of an aperiodic monotile in the translational tiling of $\mathbb{Z}^n$ for some huge dimension $n$ proved by Greenfeld and Tao. This disproves the long-standing periodic tiling conjecture. However, it is known that there is no aperiodic monotile for translational tiling of the plane. The smallest size of known aperiodic sets for translational tilings of the plane is $8$, which was discovered more than $30$ years ago by Ammann. In this paper, we prove that translational tiling of the plane with a set of $7$ polyominoes is undecidable. As a consequence of the undecidability, we have constructed a family of aperiodic sets of size $7$ for the translational tiling of the plane. This breaks the 30-year-old record of Ammann.

翻译：近期，非周期单铺砌问题在两个不同研究方向上取得了突破性成果。其一是Smith、Myers、Kaplan与Goodman-Strauss于2023年发现的平面非周期单铺砌族（允许旋转），打破了由Roger Penrose在1970年代发现的二元非周期集所保持的平面非周期集最小尺寸的50年记录。其二是Greenfeld与陶哲轩证明的$\mathbb{Z}^n$平移铺砌（在某个巨大维度$n$下）存在非周期单铺砌，这否定了长期存在的周期铺砌猜想。然而，已知平面平移铺砌不存在非周期单铺砌。目前平面平移铺砌已知非周期集的最小尺寸为$8$，由Ammann在30余年前发现。本文证明：使用7种多联骨牌进行平面平移铺砌的问题是不可判定的。基于该不可判定性，我们构造了一族尺寸为7的平面平移铺砌非周期集，从而打破了Ammann保持30年的记录。