Given a finite set $A\subset \mathbb{R}\backslash \{0\}$, define \begin{align*}&A\cdot A \ =\ \{a_i\cdot a_j\,|\, a_i,a_j\in A\},\\ &A/A \ =\ \{a_i/a_j\,|\,a_i,a_j\in A\},\\ &A + A \ =\ \{a_i + a_j\,|\, a_i,a_j\in A\},\\ &A - A \ =\ \{a_i - a_j\,|\,a_i,a_j\in A\}.\end{align*} The set $A$ is said to be MPTQ (more product than quotient) if $|A\cdot A|>|A/A|$ and MSTD (more sum than difference) if $|A + A|>|A - A|$. Since multiplication and addition are commutative while division and subtraction are not, it is natural to think that MPTQ and MSTD sets are very rare. However, they do exist. This paper first shows an efficient search for MPTQ subsets of $\{1,2,\ldots,n\}$ and proves that as $n\rightarrow \infty$, the proportion of MPTQ subsets approaches $0$. Next, we prove that MPTQ sets of positive numbers must have at least $8$ elements, while MPTQ sets of both negative and positive numbers must have at least $5$ elements. Finally, we investigate several sequences that do not have MPTQ subsets.
对于一组整数$A$，我们考虑$R(A)=\{a/b: a, b\in A, b\neq 0\}$。当$A$是由积分形式得到的非零值集时，研究$R(A)$在$p$ -adic数中的稠密性是一个公开问题.这个问题已经回答了二次形式。最近，Antony和Barman研究了对角二元三次形式$ax^3+by^3$的这个问题，其中$a$和$b$是整数。在这篇文章中，我们研究这个问题的对角线形式。我们将Antony和Barman的结果推广到对角二元形式$ax^n+by^n$的所有$n\geq 3$。我们还研究了由度$n\geq 3$的对角形式得到的非零值的导数的$p$ -adic稠密性，其中$\gcd(n,p(p-1))=1$。
Hausdorff dimension of sets with restricted, slowly growing partial quotients
I. J.Good（1941）证明了$(0,1)$中的无理数集合如果其偏导数$a_n$趋于无穷大，则它具有豪斯多夫维数$1/2$。一些相关的结果施加限制的类型$a_n\in B$或$a_n\geq f(n)$，其中$B$是一个无限子集的$\mathbb N$和$f$是一个快速增长的功能与$n$。我们证明了，对于任意的$B$和任意的$f$，其值在$[\min B,\infty)$中并趋于无穷大，$(0,1)$中的无理数集合使得\[ a_n\in B,\ a_n\leq f(n)\text{ for all $n\in\mathbb N$, and }a_n\to\infty\text{ as }n\to\infty\]具有豪斯多夫维数$τ(B)/2,$，其中$τ(B)$是$B$的收敛指数.
Product and Quotient Set Dominance in Finite Groups
University of Maryland, College Park（马里兰大学帕克分校）；University of Chicago（芝加哥大学）；Missouri State University（密苏里州立大学）；Texas A&M University（德克萨斯农工大学）；University of Rochester（罗切斯特大学）；Williams College（威廉姆斯学院）；University of Michigan（密歇根大学）
Reed College（里德学院）；University of Michigan（密歇根大学）；Amherst College（阿默斯特学院）；Princeton University（普林斯顿大学）；Williams College（威廉姆斯学院）；University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
