On new sum-product type estimates

On the Gap-sum and Gap-product Sequences of Integer Sequences

In this note, we explore two families of sequences associated to a suitable integer sequence: the gap-sum sequence and the gap-product sequence. These are the sums and the products of consecutive numbers not in the original sequence. We give closed forms for both, in terms of the original sequence, and in the case of Horadam sequences, we find the generating function of the gap-sum sequence. For some elementary sequences, we indicate that the gap-product sequences are given by the Fuss-Catalan-Raney numbers.

Sums and Products with Smooth Numbers

We estimate the sizes of the sumset A + A and the productset A $\cdot$ A in the special case that A = S (x, y), the set of positive integers n less than or equal to x, free of prime factors exceeding y.

Sum-product phenomena: p-adic case

The sum-product phenomena over a finite extension K of $\mathbb{Q}_p$ is explored. The main feature of the results is the fact that the implied constants are independent of $p$.

On popular sums and differences of sets with small products

Given a subset of real numbers $A$ with small product $AA$ we obtain a new upper bound for the additive energy of $A$. The proof uses a natural observation that level sets of convolutions of the characteristic function of $A$ have small product with $A$.

Sums and products along sparse graphs

In their seminal paper from 1983, Erdős and Szemerédi showed that any $n$ distinct integers induce either $n^{1+ε}$ distinct sums of pairs or that many distinct products, and conjectured a lower bound of $n^{2-o(1)}$. They further proposed a generalization of this problem, in which the sums and products are taken along the edges of a given graph $G$ on $n$ labeled vertices. They conjectured a version of the sum-product theorem for general graphs that have at least $n^{1+ε}$ edges. In this work, we consider sum-product theorems for sparse graphs, and show that this problem has important consequ…

On the size of $k$-fold sum and product sets of integers

We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$, such that for every finite set $A$ of integers with cardinality $|A| > 1$, we have either $$ |A + ... + A| \geq |A|^b$$ or $$ |A \cdot ... \cdot A| \geq |A|^b$$ where $A + ... + A$ and $A \cdot ... \cdot A$ are the collections of $k$-fold sums and products of elements of $A$ respectively. This is progress towards a conjecture of Erdös and Szemerédi on sum and product sets.

Partition Inequalities and Applications to Sum-Product Conjectures of Kanade-Russell

We consider differences of one- and two-variable finite products and provide combinatorial proofs of the nonnegativity of certain coefficients. Since the products may be interpreted as generating functions for certain integer partitions, this amounts to showing a partition inequality. This extends results due to Berkovich-Garvan and McLaughlin. We then apply the first inequality and Andrews' Anti-telescoping Method to give a solution to an 'Ehrenpreis Problem' for recently conjectured sum-product identities of Kanade- Russell. That is, we provide some evidence for Kanade-Russell's conjectures …

Sum-product estimates for diagonal matrices

Given $d \in \mathbb{N}$, we establish sum-product estimates for finite, non-empty subsets of $\mathbb{R}^d$. This is equivalent to a sum-product result for sets of diagonal matrices. In particular, let $A$ be a finite, non-empty set of $d \times d$ diagonal matrices with real entries. Then for all $δ_1 < 1/3 + 5/5277$, we have \[ |A+A| + |A\cdot A| \gg_{d} |A|^{1 + δ_{1}/d}. \] In this setting, the above estimate quantitatively strengthens a result of Chang.

A sum-division estimate of reals

Let $A$ be a finite set of positive real numbers. We present a sum-division estimate: |A+A|^2|A/A|\geq\frac{|A|^4}{4}.

Sum-product phenomena in F_p: a brief introduction

These notes arose from my Cambridge Part III course on Additive Combinatorics, given in Lent Term 2009. The aim was to understand the simplest proof of the Bourgain-Glibichuk-Konyagin bounds for exponential sums over subgroups. As a byproduct one obtains a clean proof of the Bourgain-Katz-Tao theorem on the sum-product phenomenon in F_p. The arguments are essentially extracted from a paper of Bourgain, and I benefitted very much from being in receipt of unpublished course notes of Elon Lindenstrauss. No originality is claimed.

On higher energy decompositions and the sum-product phenomenon

Let $A \subset \mathbb{R}$ be finite. We quantitatively improve the Balog-Wooley decomposition, that is $A$ can be partitioned into sets $B$ and $C$ such that $$\max\{E^+(B) , E^{\times}(C)\} \lesssim |A|^{3 - 7/26}, \ \ \max \{E^+(B,A) , E^{\times}(C, A) \}\lesssim |A|^{3 - 1/4}.$$ We use similar decompositions to improve upon various sum-product estimates. For instance, we show $$ |A+A| + |A A| \gtrsim |A|^{4/3 + 5/5277}.$$

Sums of Products of Bernoulli numbers of the second kind

The Bernoulli numbers b_0,b_1,b_2,.... of the second kind are defined by \sum_{n=0}^\infty b_nt^n=\frac{t}{\log(1+t)}. In this paper, we give an explicit formula for the sum \sum_{j_1+j_2+...+j_N=n, j_1,j_2,...,j_N>=0}b_{j_1}b_{j_2}...b_{j_N}. We also establish a q-analogue for \sum_{k=0}^n b_kb_{n-k}=-(n-1)b_n-(n-2)b_{n-1}.

Attaining the exponent $5/4$ for the sum-product problem in finite fields

We improve the exponent in the finite field sum-product problem from $11/9$ to $5/4$, improving the results of Rudnev, Shakan and Shkredov. That is, we show that if $A\subset \mathbb{F}_p$ has cardinality $|A|\ll p^{1/2}$ then \[ \max\{|A\pm A|,|AA|\} \gtrsim |A|^\frac54 \] and \[ \max\{|A\pm A|,|A/A|\}\gtrsim |A|^\frac54\,. \]

Sum-product estimates via directed expanders

Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset \F_q$, define $P(A):=\{P(x_1, x_2) | x_i \in A \}$. Using certain constructions of expanders, we characterize all polynomials $P$ for which the following holds \vskip2mm \centerline{\it If $|A+A|$ is small, then $|P(A)|$ is large.} \vskip2mm The case $P=x_1x_2$ corresponds to the well-known sum-product problem.

关于求和-乘积问题和凸求和集问题的注记

A Note on the Sum-Product Problem and the Convex Sumset Problem

该文为实数集上的求和-乘积猜想和凸求和集问题提供了新的指数界。

Sums and Products of Distinct Sets and Distinct Elements in $\mathbb{C}$

Let $A$ and $B$ be finite subsets of $\mathbb{C}$ such that $|B|=C|A|$. We show the following variant of the sum product phenomenon: If $|AB|<α|A|$ and $α\ll \log |A|$, then $|kA+lB|\gg |A|^k|B|^l$. This is an application of a result of Evertse, Schlickewei, and Schmidt on linear equations with variables taking values in multiplicative groups of finite rank, in combination with an earlier theorem of Ruzsa about sumsets in $\mathbb{R}^d$. As an application of the case $A=B$ we give a lower bound on $|A^+|+|A^\times|$, where $A^+$ is the set of sums of distinct elements of $A$ and $A^\times$ is …

Some remarks on the asymmetric sum--product phenomenon

Using some new observations connected to higher energies, we obtain quantitative lower bounds on $\max\{|AB|, |A+C| \}$ and $\max\{|(A+α)B|, |A+C|\}$, $α\neq 0$ in the regime when the sizes of finite subsets $A,B,C$ of a field differ significantly.

An improved sum-product estimate for general finite fields

This paper improves on a sum-product estimate obtained by Katz and Shen for subsets of a finite field whose order is not prime.

Szemeredi-Trotter type theorem and sum-product estimate in finite fields

We study a Szemerédi-Trotter type theorem in finite fields. We then use this theorem to obtain an improved sum-product estimate in finite fields.

Nonstandard proof of combined sum and product structure in IP$^{*}$ sets

V. Bergelson and N. Hindman proved that $IP^{*}$ sets contain all possible finite sum and product of a sum subsystem of any sequence in $\mathbb{N}$. In this article, we will prove this result using Nonstandard analysis.

表示论在和积现象中的若干应用

Some applications of representation theory to the sum-product phenomenon

本文提出用表示论估计关联的新方法，并将其应用于带乘法特征的和式及数论中的 Zaremba 猜想。

A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes

The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full dimensional polytopes is precisely the product of the normalized volumes of the summands.

The Sum and Product of Independence Numbers of Graphs and their Line Graphs

The bounds on the sum and product of chromatic numbers of a graph and its complement are known as Nordhaus-Gaddum inequalities. In this paper, we study the bounds on the sum and product of the independence numbers of graphs and their line graphs.

Sum-product estimates for rational functions

We establish several sum-product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)|+|AA| is substantially larger than |A| for an arbitrary polynomial f over F_p. Second, a characterization is given for the rational functions f and g for which |f(A)+f(A)|+|g(A,A)| can be as small as |A|, for large |A|. Third, we show that under mild conditions on f, |f(A,A)| is substantially larger than |A|, provided |A| is large. We also present a conjecture on what the general sum-product result should be.

Discretized sum-product estimates in matrix algebras

We generalize Bourgain's discretized sum-product theorem to matrix algebras.

The sum-product phenomenon in arbitrary rings

The \emph{sum-product phenomenon} predicts that a finite set $A$ in a ring $R$ should have either a large sumset $A+A$ or large product set $A \cdot A$ unless it is in some sense "close" to a finite subring of $R$. This phenomenon has been analysed intensively for various specific rings, notably the reals $\R$ and cyclic groups $\Z/q\Z$. In this paper we consider the problem in arbitrary rings $R$, which need not be commutative or contain a multiplicative identity. We obtain rigorous formulations of the sum-product phenomenon in such rings in the case when $A$ encounters few zero-divisors of $…

The Sum and Product of Chromatic Numbers of Graphs and their Line Graphs

A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper some variations are considered. First, recall their theorem, which gives bounds on the sum and the product of the chromatic number of a graph with that of its complement. In this paper, we provide a new characterization of the other class of graphs.

Determinant of a Sum of Certain Kronecker Products

We compute the determinant of $\sum_{n=1}^{N} \vec{A}^{(n)} \otimes \vec{B}^{(n)}$, where $\vec{A}^{(n)}$ is square and ${\vec{B}^{(n)}=\vec{x}^{(n)}{\vec{y}^{(n)}}^T}$ where $\vec{x}^{(n)}$ and $\vec{y}^{(n)}$ have length $N$.

Sum-Product Phenomena for Planar Hypercomplex Numbers

We study the sum-product problem for the planar hypercomplex numbers: the dual numbers and double numbers. These number systems are similar to the complex numbers, but it turns out that they have a very different combinatorial behavior. We identify parameters that control the behavior of these problems, and derive sum-product bounds that depend on these parameters. For the dual numbers we expose a range where the minimum value of $\max\{|A+A|,|AA|\}$ is neither close to $|A|$ nor to $|A|^2$. To obtain our main sum-product bound, we extend Elekes' sum-product technique that relies on point-line…

An improved sum-product estimate over finite fields

This paper gives an improved sum-product estimate for subsets of a finite field whose order is not prime. It is shown, under certain conditions, that $$\max\{|A+A|,|A\cdot{A}|\}\gg{\frac{|A|^{12/11}}{(\log_2|A|)^{5/11}}}.$$ This new estimate matches, up to a logarithmic factor, the current best known bound obtained over prime fields by Rudnev (\cite{mishaSP}).

Monochromatic products and sums in $2$-colorings of $\mathbb{N}$

We show that any $2$-coloring of $\mathbb{N}$ contains infinitely many monochromatic sets of the form $\{x,y,xy,x+y\},$ and more generally monochromatic sets of the form $\{x_i,\prod x_i,\sum x_i: i\leq k\}$ for any $k\in\mathbb{N}.$ Along the way we prove a monochromatic products of sums theorem that extends Hindman's theorem and a colorful variant of this result that holds in any 'balanced' coloring.

Two Kinds of Constructions of Directed Strongly Regular Graphs from Partial Sum Families and Semi-direct Products of Groups

In this paper, we construct directed strongly regular graphs with new parameters by using partial sum families with local rings. 16 families of new directed strongly regular graphs are obtained and the uniform partial sum families are given. Based on the cyclotomic numbers of finite fields, we present two infinite families of directed strongly regular Cayley graphs from semi-direct products of groups.

An application of kissing numbers in sum-product estimates

The boundedness of the kissing numbers of convex bodies has been known to Hadwiger for long. We present an application of it to the sum-product estimate $$\max(|\mathcal{A}+\mathcal{A}|,|\mathcal{A}\mathcal{A}|)\gg\dfrac{|\mathcal{A}|^{4/3}}{\lceil\log|\mathcal{A}|\rceil^{1/3}}$$ for finite sets $\mathcal{A}$ of quaternions and of a certain family of well-conditioned matrices.

An improved estimate on sums of product sets

In a recent paper \cite{Gl} A. Glibichuk proved that if $A,B$ are subsets of an arbitrary finite filed $\F_q$, such that $|A||B|>q$, then $16AB = \F_q$. We improve this to $10AB = \F_q.$

An improved sum-product bound for quaternions

We show that there exists an absolute constant $c > 0$, such that, for any finite set $A$ of quaternions, \[ \max\{|A+A, |AA| \} \gtrsim |A|^{4/3 + c}. \] This generalizes a sum-product bound for real numbers proved by Konyagin and Shkredov.

An improved sum-product inequality in fields of prime order

This note improves the best known exponent 1/12 in the prime field sum-product inequality (for small sets) to 1/11, modulo a logarithmic factor.

由指数生成函数的 Cauchy 乘积得到的 $r$-错排求和规则

A sum rule for $r$-derangements obtained from the Cauchy product of exponential generating functions

本文提出涉及二项式系数的 $r$-错排求和规则，利用两个指数生成函数的 Cauchy 乘积证明该恒等式，推广了常规错排的已知关系。

并行中的无限和积配置

Infinite Sum-Product Configurations in Parallel

本文证明了对于自然数的任意有限划分，存在一个无限序列，其有限和是单色的，且固定因子数的无限多个乘积也是单色的，并应用此技术证明多种复杂无限和积配置在任意有限着色下必为单色。

乘积与和节流数的精确界

Sharp bounds for product and sum throttling numbers

本研究提出新技术，为幂支配的和节流与初始成本乘积节流建立图阶数相关的精确上界，同时确定了三类图问题中特定图操作对乘积节流数变化影响的精确界。

Sum-Product Type Estimates for Subsets of Finite Valuation Rings

Let $R$ be a finite valuation ring of order $q^r.$ Using a point-plane incidence estimate in $R^3$, we obtain sum-product type estimates for subsets of $R$. In particular, we prove that for $A\subset R$, $$|AA+A|\gg \min\left\{q^{r}, \frac{|A|^3}{q^{2r-1}}\right\}.$$ We also show that if $|A+A||A|^{2}>q^{3r-1}$, then $$|A^2+A^2||A+A|\gg q^{\frac{r}{2}}|A|^{\frac{3}{2}}.$$

Variations on the Sum-Product Problem

This paper considers various formulations of the sum-product problem. It is shown that, for a finite set $A\subset{\mathbb{R}}$, $$|A(A+A)|\gg{|A|^{\frac{3}{2}+\frac{1}{178}}},$$ giving a partial answer to a conjecture of Balog. In a similar spirit, it is established that $$|A(A+A+A+A)|\gg{\frac{|A|^2}{\log{|A|}}},$$ a bound which is optimal up to constant and logarithmic factors. We also prove several new results concerning sum-product estimates and expanders, for example, showing that $$|A(A+a)|\gg{|A|^{3/2}}$$ holds for a typical element of $A$.

Sum-product estimates in finite fields

We prove, using combinatorics and Kloosterman sum technology that if $A \subset {\Bbb F}_q$, a finite field with $q$ elements, and $q^{1/2} \lesssim |A| \lesssim q^{7/10}$, then $\max \{|A+A|, |A \cdot A|\} \gtrsim \frac{{|A|}^{3/2}}{q^{1/4}$.

A Sum-Product Estimate for Well Spaced Sets

We study the $δ$-discretized sum-product estimates for well spaced sets. Our main result is: for a fixed $α\in(1,\frac{3}{2}]$, we prove that for any $\sim|A|^{-1}$-separated set $A\subset[1,2]$ and $δ=|A|^{-α}$, we have: $\mathcal{N}(A+A, δ)\cdot \mathcal{N}(AA, δ) \gtrsim_ε|A|δ^{-1+ε}$.

An Algorithm to Solve the Equal-Sum-Product Problem

A recursive algorithm is constructed which finds all solutions to a class of Diophantine equations connected to the problem of determining ordered n-tuples of positive integers satisfying the property that their sum is equal to their product. An examination of the use of Binary Search Trees in implementing the algorithm into a working program is given. In addition an application of the algorithm for searching possible extra exceptional values of the equal-sum-product problem is explored after demonstrating a link between these numbers and the Sophie Germain primes.

Ahlfors正则集的和积现象

Sum-product phenomena for Ahlfors-regular sets

该研究利用Orponen的成果，得到Ahlfors正则集的和积结果，进而获得Solymosi 4/3界的分形类似结论。

Discretized Sum-product and Fourier decay in $\mathbb{R}^n$

We generalize Bourgain's theorem on the decay of the Fourier transform of the multiplicative convolution of measures on $\mathbb R$ to the ring $\mathbb R^n$, where the multiplication is given by coordinate multiplication.

On a $ψ_1$ - norm estimate of sum of dependent random variables using simple random walk on graph

We obtained a $ψ_1$ estimate for the sum of Rademacher random variables under condition that they are dependent.

关于分形集求和-乘积问题的注记

A note on the sum-product problem for fractal sets

研究分形集的求和-乘积问题，利用几何与离散求和-乘积理论，证明分形集的乘积集或和集的盒维数下界，改进后的差集下界更优。

两个半正定算子张量积之和的优超关系

A majorization relation for a sum of two tensor products of positive semidefinite operators

研究两个半正定算子张量积之和的Ky Fan优超关系，用线性规划证明其可分版本，还通过例子表明三个或更多此类张量积之和该关系一般不成立。