Partition regularity for systems of diagonal equations

虚二次整数环中的分拆正则性

Partition regularity in imaginary quadratic rings of integers

本文研究了具有良好线性因子分解的二次 homogeneous 方程在虚二次整数环中的分拆正则性，提出了数论结果，如无周期完全乘法函数的特征化、Turán-Kubilius 不等式和乘法函数的新集中估计。

4对角双移位平面分割的麦克马洪分析视角

A MacMahon Analysis View of 4 Diagonal DSPPs

利用麦克马洪分割分析研究具有三元轮廓的斜双移位平面分割，给出新生成函数公式，与相关恒等式有关，还研究了q级数恒等式层次结构及生成函数回文性质，从线性分割角度考察得到新恒等式。

Pythagorean Partition-Regularity and Ordered Triple Systems with the Sum Property

Is it possible to color the naturals with finitely many colors so that no Pythagorean triple is monochromatic? This question is even open for two colors. A natural strategy is to show that some small nonbipartite triple systems cannot be realized as a family of Pythagorean triples. It suffices to consider partial triple systems (PTS's), and it is therefore natural to consider the Fano plane, the smallest nonbipartite PTS. We show that the Pythagorean triples do not contain any Fano plane. In fact, our main result is that a much larger family of "ordered" triple systems (viz. those with a certa…

齐次二次型的分划正则性：当前趋势与挑战

Partition regularity of homogeneous quadratics: Current trends and challenges

本文介绍算术拉姆齐理论中齐次二次型分划正则性的近期进展，涵盖基本问题、遍历理论方法及未来挑战。

Uniformity of multiplicative functions and partition regularity of some quadratic equations

Since the theorems of Schur and van der Waerden, numerous partition regularity results have been proved for linear equations, but progress has been scarce for non-linear ones, the hardest case being equations in three variables. We prove partition regularity for certain equations involving forms in three variables, showing for example that the equations $16x^2+9y^2=n^2$ and $x^2+y^2-xy=n^2$ are partition regular, where $n$ is allowed to vary freely in $\mathbb{N}$. For each such problem we establish a density analogue that can be formulated in ergodic terms as a recurrence property for actions…

广义毕达哥拉斯对的划分正则性

Partition regularity of generalized Pythagorean pairs

研究齐次二次方程划分正则性，在系数自然条件下，对正整数有限染色，存在\(ax^2 + by^2 = cz^2\)解使\(x,y\)等颜色相同，部分结果依赖猜想，证明基于扩展先前论证，利用乘法函数性质。

Matricial formulae for partitions

The exponential of the triangular matrix whose entries in the diagonal at distance $n$ from the principal diagonal are all equal to the sum of the inverse of the divisors of $n$ is the triangular matrix whose entries in the diagonal at distance $n$ from the principal diagonal are all equal to the number of partitions of $n$. A similar result is true for any pair of sequences satisfying a special recurrence.

对角有理分拆的算术约束与极限定律

Arithmetic Constraints and Limit Laws for Diagonal Rational Partitions

本文研究对角有理分拆的计数渐近与极限分布，给出主项及修正项，并证明素数分母块总和的阈值行为及残差分布的高斯极限。

On Rado conditions for nonlinear Diophantine equations

Building on previous work of Di Nasso and Luperi Baglini, we provide general necessary conditions for a Diophantine equation to be partition regular. These conditions are inspired by Rado's characterization of partition regular linear homogeneous equations. We conjecture that these conditions are also sufficient for partition regularity, at least for equations whose corresponding monovariate polynomial is linear. This would provide a natural generalization of Rado's theorem. We verify that such a conjecture hold for the equations $x^{2}-xy+ax+by+cz=0$ and $x^{2}-y^{2}+ax+by+cz=0$ for $a,b,c\in…

Counting monochromatic solutions to diagonal Diophantine equations

We show how to adapt the Hardy--Littlewood circle method to count monochromatic solutions to diagonal Diophantine equations. This delivers a lower bound which is optimal up to absolute constants. The method is illustrated on equations obtained by setting a diagonal quadratic form equal to a linear form. As a consequence, we determine an algebraic criterion for when such equations are partition regular. Our methods involve discrete harmonic analysis and require a number of `mixed' restriction estimates, which may be of independent interest.

单色和与商在自然数中的研究

Monochromatic sums and quotients in $\mathbb N$

本文证明了配置x,y,x+y,y/x在强无穷形式下的分拆正则性，扩展了Hindman定理，并研究了涉及x和y一次多项式乘积的分拆正则性问题。

Regularity versus complexity in the binary representation of 3^n

We use the grid consisting of bits of 3^n to motivate the definition of 2-adic numbers. Specifically, we exhibit diagonal stripes in the bits of 3^(2^n), which turn out to be the first in an infinite sequence of such structures. Our observations are explained by a 2-adic power series, providing some regularity among the disorder in the bits of powers of 3. Generally, the base-p representation of k^(p^n) has these features.

On a class of diagonal equations over finite fields

Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form $x_1^{2^m}+\dots+x_n^{2^m}=0$ over a finite field of characteristic $p\equiv\pm 3\pmod{8}$. All of the evaluations are effected in terms of parameters occurring in quadratic partitions of some powers of $p$.