On sumfree subsets of hypercubes

Classifications of $\ell$-Zero-Sumfree Sets

The set of all $\ell$-zero-sumfree subsets of $\mathbb{Z}/n\mathbb{Z}$ is a simplicial complex denoted by $Δ_{n,\ell}$ We create an algorithm via defining a set of integer partitions we call $(n,\ell)$-congruent partitions in order to compute this complex for moderately-sized parameters $n$ and $\ell$. We also theoretically determine $Δ_{n,\ell}$ for several infinite families of parameters, and compute the intersection lattices and the characteristic polynomials of the corresponding coordinate subspace arrangements.

The Schur degree of additive sets

Let (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x +y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The classical inequality S(n) $\le$ R n (3) -- 2, between the Schur number S(n) and the Ramsey number R n (3) = R(3,. .. , 3), is shown to remain valid in a wider context, involving the Schur degree of certain subsets of G. Recursive upper bounds are known for R n (3) but not for S(n) so far. We formulate a conjecture which, if true, would fill this gap. Indeed, our s…

On the Olson and the Strong Davenport constants

A subset $S$ of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of $S$ is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, $p$-groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general results, establishing new lower bounds for the cardinality of zero-sumfree sets for various …

关于格立方体中最大无和子集

On the largest sum-free subset of the lattice cube

本文确定了所有维度d下格立方体{1,2,…,n}^d中最大无和子集的极限密度，证明其由两个超平面切片构成，并给出反例说明对不含原点的凸集该结论不成立。

Galois 轨道中扭点在近非环面集附近多胞体上的分布

Galois orbits of torsion points over polytopes near atoral sets

该研究推广了扭点 Galois 轨道上的等分布定理至多胞体子集，给出收敛速度估计，并应用于二维高度收敛问题，附有计算严格度幂的算法。

具有受限差集的Sperner系统

Sperner systems with restricted differences

本文研究了q-模L差集Sperner系统，利用p-进分析和多项式方法获得新的上界，扩展并改进了现有结果，并首次在q-模设定中提出Snevily定理的类比。