Previously, the author introduced quasirandom permutations, permutations of $\mathbb{Z}_n$ which map intervals to sets with low discrepancy. Here we show that several natural number-theoretic permutations are quasirandom, some very strongly so. Quasirandomness is established via discrete Fourier analysis and the Erdos-Turan inequality, as well as by other means. We apply our results on Sos permutations to make progress on a number of questions relating to the sequence of fractional parts of multiples of an irrational. Several intriguing new open problems are presented throughout the discussion.
Let $S_{\rm lcm}(n)$ denote the set of permutations $π$ of $[n]=\{1,2,\dots,n\}$ such that ${\rm lcm}[j,π(j)]\le n$ for each $j\in[n]$. Further, let $S_{\rm div}(n)$ denote the number of permutations $π$ of $[n]$ such that $j\midπ(j)$ or $π(j)\mid j$ for each $j\in[n]$. Clearly $S_{\rm div}(n)\subset S_{\rm lcm}(n)$. We get upper and lower bounds for the counts of these sets, showing they grow geometrically. We also prove a conjecture from a recent paper on the number of "anti-coprime" permutations of $[n]$, meaning that each $\gcd(j,π(j))>1$ except when $j=1$.
Simultaneous Arithmetic Progressions on Algebraic Curves
一个同时的等差数列（s. a. p.）由k个点（x_i，y_σ（i））组成，其中x_i，y_i是等差数列，σ是置换。Garcia-Selfa和Tornero问是否有一个长度上的s. a. p.的椭圆曲线在维尔斯特拉斯形式超过Q。我们证明了4319是R上曲线的这样一个界。这是通过将网格中曲线的平移视为图形来完成的。一个简单的上界的交叉数和“交叉不等式”给出了一个下界。这些因素共同决定了曲线上s. a. p.的长度。然后，我们使用类似的方法将结果推广到任意实代数曲线。而不是考虑s. a. p.我们考虑网格中的k^2/3点。交叉点的数目由Bezout定理限定。然后，我们给出了另一个证明使用的结果Jarnik界定的数量网格点的凸曲线。这个结果适用于任何实代数曲线可以分为凸和凹的部分，其数量取决于程度。最后，将这些结果推广到复代数曲线。
In this paper we discuss the permutational property of polynomials of the form $f(L(x))+k(L(x))\cdot M(x)\in \mathbb F_{q^n}[x]$ over the finite field $\mathbb F_{q^n}$, where $L, M\in \mathbb F_q[x]$ are $q$-linearized polynomials. The restriction $L, M\in \mathbb F_q[x]$ implies a nice correspondence between the pair $(L, M)$ and the pair $(g, h)$ of conventional $q$-associates over $\mathbb F_q$ of degree at most $n-1$. In particular, by using the AGW criterion, permutational properties of our class of polynomials translates to some arithmetic properties of polynomials over $\mathbb F_q$, l…
