Melvyn B. Nathanson
An open problem about finite geometric progressions in syndetic sets leads to a family of diophantine equations related to the commutativity of translation and multiplication by squares.
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In order to investigate multiplicative structures in additively large sets, Beiglböck et al. raised a significant open question as to whether or not every subset of the natural numbers with bounded gaps (syndetic set) contains arbitrarily long geometric progressions. A result of Erdős implies that syndetic sets contain a $2$-term geometric progression with integer common ratio, but we still do not know if they contain such a progression with common ratio being a perfect square. In this article, we prove that for each $k\in \mathbb{N}$, a syndetic set contains $2$-term geometric progressions wi…
We show that there exists a positive constant C such that the following holds: Given an infinite arithmetic progression A of real numbers and a sufficiently large integer n (depending on A), there needs at least Cn geometric progressions to cover the first n terms of A. A similar result is presented, with the role of arithmetic and geometric progressions reversed.
Circles with four rational points in geometric progression
本文证明了存在无穷多个具有有理半径的圆，其上含有四个横坐标或纵坐标成等比数列的有理点，推广了已知的三个点的情形。
The law of the iterated logarithm for discrepancies of geometric progressions with small ratios is proved.
We establish a transference inequality conjectured by Badziahin and Bugeaud relating exponents of rational approximation of points in geometric progression.
We prove the law of the iterated logarithm for discrepancies of the sequence $\{(3/2)^kx\}$.
A sequence of rational points on an algebraic planar curve is said to form an $r$-geometric progression sequence if either the abscissae or the ordinates of these points form a geometric progression sequence with ratio $r$. In this work, we prove the existence of infinitely many rational numbers $r$ such that for each $r$ there exist infinitely many $r$-geometric progression sequences on the unit circle $x^2 + y^2 = 1$ of length at least $3$.
Arithmetic progressions in sumsets of geometric progressions
本文研究了几何数列和集中的等差数列结构，结合线性形式对数界和模ularity 性质，给出完整刻画方法。
Let $C$ be a hyperelliptic curve over $\mathbb Q$ described by $y^2=a_0x^n+a_1x^{n-1}+\ldots+a_n$, $a_i\in\mathbb Q$. The points $P_{i}=(x_{i},y_{i})\in C(\mathbb{Q})$, $i=1,2,...,k,$ are said to be in a geometric progression of length $k$ if the rational numbers $x_{i}$, $i=1,2,...,k,$ form a geometric progression sequence in $\mathbb Q$, i.e., $x_i=pt^{i}$ for some $p,t\in\mathbb Q$. In this paper we prove the existence of an infinite family of hyperelliptic curves on which there is a sequence of rational points in a geometric progression of length at least eight.
Geometric Progressions meet Zeckendorf Representations
本文研究了在Zeckendorf数制中几何级数与禁止二进制模式的交互问题，证明了相关指数集合的有限性或周期性。
A geometric progression of length $k$ and integer ratio is a set of numbers of the form $\{a,ar,\dots,ar^{k-1}\}$ for some positive real number $a$ and integer $r\geq 2$. For each integer $k \geq 3$, a greedy algorithm is used to construct a strictly decreasing sequence $(a_i)_{i=1}^{\infty}$ of positive real numbers with $a_1 = 1$ such that the set \[ G^{(k)} = \bigcup_{i=1}^{\infty} \left(a_{2i} , a_{2i-1} \right] \] contains no geometric progression of length $k$ and integer ratio. Moreover, $G^{(k)}$ is a maximal subset of $(0,1]$ that contains no geometric progression of length $k$ and in…
Let s and k be integers with s \geq 2 and k \geq 2. Let g_k^{(s)}(n) denote the cardinality of the largest subset of the set {1,2,..., n} that contains no geometric progression of length k whose common ratio is a power of s. Let r_k(\ell) denote the cardinality of the largest subset of the set {0,1,2,\ldots, \ell -1\} that contains no arithmetric progression of length k. The limit \[ \lim_{n\rightarrow \infty} \frac{g_k^{(s)}(n)}{n} = (s-1) \sum_{m=1}^{\infty} \left(\frac{1}{s} \right)^{\min \left(r_k^{-1}(m)\right)} \] exists and converges to an irrational number.
A geometric-arithmetic progression of primes is a set of $k$ primes (denoted by GAP-$k$) of the form $p_1 r^j + j d$ for fixed $p_1$, $r$ and $d$ and consecutive $j$, {\it i.e}, $\{p_1, \, p_1 r + d, \, p_1 r^2 + 2 d, \, p_1 r^3 + 3 d, \,...}$. We study the conditions under which, for $k \ge 2$, a GAP-$k$ is a set of $k$ primes in geometric-arithmetic progression. Computational data (along with the MATHEMATICA codes) containing progressions up to GAP-13 is presented. Integer sequences for the sets of differences $d$ corresponding to the GAPs of orders up to 11 are also presented.
We present a certain generalization of a recent result of M. I. Cirnu on linear recurrence relations with coefficient in progressions [2]. We provide some interesting examples related to some well-known integer sequences, such as Fibonacci sequence, Pell sequence, Jacobsthal sequence, and the Balancing sequence of numbers. The paper also provides several approaches in solving the linear recurrence relation under consideration. We end the paper by giving out an open problem.
Koksma's equidistribution theorem from 1935 states that for Lebesgue almost every $α>1$, the fractional parts of the geometric progression $(α^{n})_{n\geq1}$ are equidistributed modulo one. In the present paper we sharpen this result by showing that for almost every $α>1$, the correlations of all finite orders and hence the normalized gaps of $(α^{n})_{n\geq1}$ mod 1 have a Poissonian limit distribution, thereby resolving a conjecture of the two first named authors. While an earlier approach used probabilistic methods in the form of martingale approximation, our reasoning in the present paper …
We investigate the problem of the distribution of sums of functions of prime numbers located on an arithmetic progression. This problem is closely related to the problem of the distribution of prime numbers on an arithmetic progression. Based on this distribution, a general formula was obtained for the asymptotic estimate of the sums of functions of primes, and also asymptotics was found for the sums of various functions of primes on a geometric progression. Several assertions about asymptotics of sums of functions of prime numbers on a geometric progression are proved. Necessary and sufficien…
Two well studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression, or in geometric progression. We study generalizations of this problem, by varying the kinds of progressions to be avoided and the metrics used to evaluate the density of the resulting subsets. One can view a 3-term arithmetic progression as a sequence $x, f_n(x), f_n(f_n(x))$, where $f_n(x) = x + n$, $n$ a nonzero integer. Thus avoiding three-term arithmetic progressions is equivalent to containing no three elements of the form $x, f_n(x), f_n(f_n…
By a classical theorem of Koksma the sequence of fractional parts $(\{x^n\})_{n \geq 1}$ is uniformly distributed for almost all values of $x$. In the present paper we obtain an exact quantitative version of Koksma's theorem, by calculating the precise asymptotic order of the discrepancy of $(\{ξx^{s_n}\})_{n \geq 1}$ for typical values of $x>1$ (in the sense of Lebesgue measure). Here $ξ>0$ is an arbitrary constant, and $(s_n)_{n \geq 1}$ can be any increasing sequence of positive integers.
An improved upper bound is obtained for the density of sequences of positive integers that contain no k-term geometric progression.
We study the arithmetic (geometric) progressions in the $x$-coordinates of quadratic points on smooth projective planar curves defined over a number field $k$. Unless the curve is hyperelliptic, we prove that these progressions must be finite. We, moreover, show that the arithmetic gonality of the curve determines the infinitude of these progressions in the set of $\overline{k}$-points with field of definition of degree at most $n$, $n\ge 3$.
Solutions of the Markoff-Rosenberger equation ax^2+by^2+cz^2 = dxyz such that their coordinates belong to the ring of integers of a number field and form a geometric progression are studied.
The problem of looking for subsets of the natural numbers which contain no 3-term arithmetic progressions has a rich history. Roth's theorem famously shows that any such subset cannot have positive upper density. In contrast, Rankin in 1960 suggested looking at subsets without three-term geometric progressions, and constructed such a subset with density about 0.719. More recently, several authors have found upper bounds for the upper density of such sets. We significantly improve upon these bounds, and demonstrate a method of constructing sets with a greater upper density than Rankin's set. Th…
Given two sets $\cA, \cB \subseteq \F_q$ of elements of the finite field $\F_q$ of $q$ elements, we show that the productset $$ \cA\cB = \{ab | a \in \cA, b \in\cB\} $$ contains an arithmetic progression of length $k \ge 3$ provided that $k<p$, where $p$ is the characteristic of $\F_q$, and $# \cA # \cB \ge 3q^{2d-2/k}$. We also consider geometric progressions in a shifted productset $\cA\cB +h$, for $f \in \F_q$, and obtain a similar result.
Several recent papers have considered the Ramsey-theoretic problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to number fields, determining bounds on the greatest possible density of ideals avoiding geometric progressions. We study the analogous problem over $\mathbb{F}_q[x]$, first constructing a set greedily which avoids these progressions and calculating its density, and then considering bounds on the upper density of subsets of $\mathbb{F}_q[x]$ which avoid 3-term geometric progressions. Thi…
We prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of prime numbers. The proof uses the probabilistic method. Using the same techniques we improve the bounds obtained by He for gaps in geometric-progression-free sets.
Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions
本文将不含三项等比数列的最大密度问题推广到非交换的 Hurwitz 四元数序，给出贪心集合密度的精确公式，并推导最优上密度的上下界。
A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets "greedily," a generalization of…
Various authors, including McNew, Nathanson and O'Bryant, have recently studied the maximal asymptotic density of a geometric progression free sequence of positive integers. In this paper we prove the existence of geometric progression free sequences with small gaps, partially answering a question posed originally by Beiglböck et al. Using probabilistic methods we prove the existence of a sequence $T$ not containing any $6$-term geometric progressions such that for any $x\geq1$ and $\varepsilon>0$ the interval $[x,x+C_{\varepsilon}\exp((C+\varepsilon)\log x/\log\log x)]$ contains an element of…
Nonlinear polynomial selection algorithms for the number field sieve address the problem of constructing polynomials with small coefficients by reducing to instances of the well-studied problem of finding short vectors in lattices. The reduction rests upon the construction of modular geometric progressions with small terms. In this paper, the methods used to construct the geometric progressions are extended, resulting in the development of two nonlinear polynomial selection algorithms.
In some of his final papers, V.I. Arnold studied pseudorandomness properties of finite deterministic sequences, which he measured in terms of their "stochasticity parameter". In the present paper we illustrate the background in probability theory and number theory of some of his considerations, and give answers to some of the questions raised in his papers.
When $k$ is a constant at least $3$, a sequence $S$ of positive integers is called $k$-GP-free if it contains no nontrivial $k$-term geometric progressions. Beiglbök, Bergelson, Hindman and Strauss first studied the existence of a $ $$k$-GP-free sequence with bounded gaps. In a previous paper the author gave a partial answer to this question by constructing a $6$-GP-free sequence $S$ with gaps of size $O(\exp(6\log n/\log\log n))$. We generalize this problem to allow the gap function $k$ to grow to infinity, and ask: for which pairs of functions $(h,k)$ do there exist $k$-GP-free sequences wit…
Geometric progressions in the sets of values of rational functions
本文研究有理函数值集中是否包含完整等比数列的问题，对一类齐次型次数差为1的有理函数给出充要条件，并通过椭圆曲线有理点对特定函数进行数值实验，提出若干猜想。
Alternating geometric progressions modulo one and Sturmian words
研究了模一的交替几何级数中，通过斯特姆词描述所有无理数ξ，使得序列(ξ(-b)^n)的像包含在长度为b^{-1}+b^{-2}-b^{-3}的区间内。
The number field sieve is the most efficient known algorithm for factoring large integers that are free of small prime factors. For the polynomial selection stage of the algorithm, Montgomery proposed a method of generating polynomials which relies on the construction of small modular geometric progressions. Montgomery's method is analysed in this paper and the existence of suitable geometric progressions is considered.
Van der Corput and metric theorems for geometric progressions for self-similar measures
本文证明了非原子自相似测度的 van der Corput 引理，并以此证明 $( x^n \mod 1 )$ 的有限阶相关函数对几乎处处 $x$ 收敛于 Poisson 模型，同时完善了关于非仿射实解析 IFS 自共形测度具有多项式 Fourier 衰减的结论。
Motzkin posed the problem of finding the maximal density $μ(M)$ of sets of integers in which the differences given by a set $M$ do not occur. The problem is already settled when $|M|\leq 2$ and $M$ is a finite arithmetic progression. In this paper, we determine $μ(M)$ when $M$ has some other structure. For example, we determine $μ(M)$ when $M$ is a finite geometric progression.
The sum-product problem for small sets II
本文研究了小集的求和-乘积问题，证明了特定大小的自然数集合具有至少30个不同的求和或乘积，并给出了最优示例。
We consider the multiplicative structure of sets of the form AA+1, where where A is a large, finite set of real numbers. In particular, we show that the additively shifted product set, AA+1 must have a large part outside of any generalized geometric progression of comparable length. We prove an analogous result in finite fields as well.
Parity of the partition function in quadratic progressions
本文证明了2010年提出的关于特定二次级数中分拆函数奇偶性包含无穷多奇偶项的猜想，通过几何方法结合伽罗瓦表示等完成证明，相关恒等式已在Lean中验证。
Let $Q(n)$ denote the count of the primitive subsets of the integers $\{1,2\ldots n\}$. We give a new proof that $Q(n) = α^{(1+o(1))n}$ which allows us to give a good error term and to improve upon the lower bound for the value of this constant $α$. We also show that the method developed can be applied to many similar problems that can be stated in terms of the divisor graph, including other questions about primitive sets, geometric-progression-free sets, and the divisor graph path-cover problem.
Let G be an arbitrary finite group and let S and T be two subsets such that |S|>1, |T|>1, and |TS|< |T|+|S|< |G|-1. We show that if |S|< |G|-4|G|^{1/2}+1 then either S is a geometric progression or there exists a non-trivial subgroup H such that either |HS|< |S|+|H| or |SH| < |S|+|H|. This extends to the nonabelian case classical results for Abelian groups. When we remove the hypothesis |S|<|G|-4|G|^{1/2}+1 we show the existence of counterexamples to the above characterization whose structure is described precisely.
Realising sets of integers as mapping degree sets
本文研究了闭定向流形间映射度集合 $D(M,N)$ 的可实现性，证明了存在不可实现的无穷整数集，同时指出包含零的有限等差数列和从1开始的正整数有限等比数列均可被实现。
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