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Register a new userTame automorphisms with multidegrees in the form of arithmetic progressions
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Let $(a,a+d,a+2d)$ be an arithmetic progression of positive integers. The following statements are proved: (1) If $amid 2d$, then $(a, a+d, a+2d)inmdeg(Tame(mathbb{C}^3))$. (2) If $a mid 2d$, then, except for arithmetic progressions of the form $(4i,4i+ij,4i+2ij)$ with $i,j inmathbb{N}$ and $j$ is an odd number, $(a, a+d, a+2d) otinmdeg(Tame(mathbb{C}^3))$. We also related the exceptional unknown case to a conjecture of Jie-tai Yu, which concerns with the lower bound of the degree of the Poisson bracket of two polynomials.
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Let $3leq d_1leq d_2leq d_3$ be integers. We show the following results: (1) If $d_2$ is a prime number and $frac{d_1}{gcd(d_1,d_3)} eq2$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_1=d_2$ or $d_3in d_1mathbb{N}+d_2mathbb{N}$; (2) If $d_3$ is a prime number and $gcd(d_1,d_2)=1$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_3in d_1mathbb{N}+d_2mathbb{N}$. We also relate this investigation with a conjecture of Drensky and Yu, which concerns with the lower bound of the degree of the Poisson bracket of two polynomials, and we give a counter-example to this conjecture.
In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers $n$ and $k$, the expression $aw([n],k)$ denotes the smallest number of colors with which the integers ${1,ldots,n}$ can be colored and still guarantee there is a rainbow arithmetic progression of length $k$. We establish that $aw([n],3)=Theta(log n)$ and $aw([n],k)=n^{1-o(1)}$ for $kgeq 4$. For positive integers $n$ and $k$, the expression $aw(Z_n,k)$ denotes the smallest number of colors with which elements of the cyclic group of order $n$ can be colored and still guarantee there is a rainbow arithmetic progression of length $k$. In this setting, arithmetic progressions can wrap around, and $aw(Z_n,3)$ behaves quite differently from $aw([n],3)$, depending on the divisibility of $n$. As shown in [Jungic et al., textit{Combin. Probab. Comput.}, 2003], $aw(Z_{2^m},3) = 3$ for any positive integer $m$. We establish that $aw(Z_n,3)$ can be computed from knowledge of $aw(Z_p,3)$ for all of the prime factors $p$ of $n$. However, for $kgeq 4$, the behavior is similar to the previous case, that is, $aw(Z_n,k)=n^{1-o(1)}$.
Celebrated theorems of Roth and of Matouv{s}ek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $Theta(n^{1/4})$. We study the analogous problem in the $mathbb{Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $mathbb{Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $mathbb{Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $mathbb{Z}_n$ is $Theta(n^{1/3+r_k/(6k)})$, where $r_k in {0,1,2}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav.
In this note we are interested in the problem of whether or not every increasing sequence of positive integers $x_1x_2x_3...$ with bounded gaps must contain a double 3-term arithmetic progression, i.e., three terms $x_i$, $x_j$, and $x_k$ such that $i + k = 2j$ and $x_i + x_k = 2x_j$. We consider a few variations of the problem, discuss some related properties of double arithmetic progressions, and present several results obtained by using RamseyScript, a high-level scripting language.
We study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple Dirichlet series has meromorphic continuation to $mathbb{C}^2$ and use Tauberian methods to obtain counts for arithmetic progressions of squares and rational points on $x^2+y^2=2$.
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