Sum-product Problem Research Articles

On the discretised 𝐴𝐵𝐶 sum-product problem

Let 0 > β ≤ α > 1 0 > \beta \leq \alpha > 1 and κ > 0 \kappa > 0 . I prove that there exists η > 0 \eta > 0 such that the following holds for every pair of Borel sets A , B ⊂ R A,B \subset \mathbb {R} with dim H ⁡ A = α \dim _{\mathrm {H}} A = \alpha and dim H ⁡ B = β \dim _{\mathrm {H}} B = \beta : dim H ⁡ { c ∈ R : dim H ⁡ ( A + c B ) ≤ α + η } ≤ α − β 1 − β + κ . \begin{equation*} \dim _{\mathrm {H}} \{c \in \mathbb {R}: \dim _{\mathrm {H}} (A + cB) \leq \alpha + \eta \} \leq \tfrac {\alpha - \beta }{1 - \beta } + \kappa . \end{equation*} This extends a result of Bourgain from 2010, which contained the case α = β \alpha = \beta . The paper also contains a δ \delta -discretised, and somewhat stronger, version of the estimate above, and new information on the size of long sums of the form a 1 B + … + a n B a_{1}B + \ldots + a_{n}B .

On a Polynomial Version of the Sum-Product Problem for Subgroups

On a Polynomial Version of the Sum-Product Problem for Subgroups

On a Polynomial Version of the Sum-Product Problem for Subgroups

Мы обобщаем два результата работ [1], [2] о суммах подмножеств $\mathbb{F}_p$ на более общую ситуацию, когда вместо суммы $x+y$ рассматривается величина $P(x,y)$, где $P$ - многочлен достаточно общего вида. В частности, получена нижняя оценка мощности множества значений многочлена $P(x,y)$, где переменные $x$ и $y$ принадлежат подгруппе $G$ мультипликативной группа поля $\mathbb{F}_p$. Также мы доказываем, что если подгруппа $G$ может быть представлена как множество значений многочлена $P(x,y)$ при $x\in A$, $y\in B$, то мощности множеств $A$ и $B$ по порядку близки к $\sqrt{|G|}$ . Библиография: 7 названий.

On Sum Sets and Convex Functions

In this paper we prove new bounds for sums of convex or concave functions. Specifically, we prove that for all $A,B \subseteq \mathbb R$ finite sets, and for all $f,g$ convex or concave functions, we have $$|A + B|^{38}|f(A) + g(B)|^{38} \gtrsim |A|^{49}|B|^{49}.$$
 This result can be used to obtain bounds on a number of two-variable expanders of interest, as well as to the asymmetric sum-product problem. We also adjust our technique to prove the three-variable expansion result $$|AB+A|\gtrsim |A|^{\frac{3}{2} +\frac{3}{170}}\,.$$Our methods follow a series of recent developments in the sum-product literature, presenting a unified picture. Of particular interest is an adaptation of a regularisation technique of Xue, originating in a paper of Rudnev, Shakan, and Shkredov, that enables us to find positive proportion subsets with certain desirable properties.

Some sum-product estimates in matrix rings over finite fields

We study some sum-product problems over matrix rings. Firstly, for A,B,C⊆Mn(Fq), we have|A+BC|≳qn2, whenever |A||B||C|≳q3n2−n+12. Secondly, if a set A in Mn(Fq) satisfies |A|≥C(n)qn2−1 for some sufficiently large C(n), then we havemax⁡{|A+A|,|AA|}≳min⁡{|A|2qn2−n+14,qn2/3|A|2/3}. These improve the results due to The and Vinh (2020), and generalize the results due to Mohammadi, Pham, and Wang (2021). We also give a new proof for a recent result due to The and Vinh (2020). Our method is based on spectral graph theory and linear algebra.

Attaining the Exponent 5/4 for the Sum-Product Problem in Finite Fields

AbstractWe improve the exponent in the finite field sum-product problem from $11/9$ to $5/4$, improving the results of Rudnev et al. [16]. 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|^\frac 54 $ and $ \max \{|A\pm A|,|A/A|\}\gtrsim |A|^\frac 54 $.

Asymmetric estimates and the sum-product problems

We show two asymmetric estimates, one on the number of collinear triples and the other on that of solutions to $(a_1+a_2)(a_1^{\prime \prime \prime }+a_2^{\prime \prime \prime })=(a_1^\prime +a_2^\prime )(a_1^{\prime \prime }+a_2^{\prime \prime })$. As ap

Semiring programming: A semantic framework for generalized sum product problems

To solve hard problems, AI relies on a variety of disciplines such as logic, probabilistic reasoning, machine learning and mathematical programming. Although it is widely accepted that solving real-world problems requires an integration amongst these, contemporary representation methodologies offer little support for this.In an attempt to alleviate this situation, we position and motivate a new declarative programming framework in this paper. We focus on the semantical foundations in service of providing abstractions of well-known problems such as SAT, Bayesian inference, generative models, learning and convex optimization. Programs are understood in terms of first-order logic structures with semiring labels, which allows us to freely combine and integrate problems from different AI disciplines and represent non-standard problems over unbounded domains. Thus, the main thrust of this paper is to view such well-known problems through a unified lens in the hope that appropriate solver strategies (exact, approximate, portfolio or hybrid) may emerge that tackle real-world problems in a principled way.

ПОЛИНОМИАЛЬНЫЙ ВАРИАНТ ЗАДАЧИ СУММ-ПРОИЗВЕДЕНИЙ

This work is about the generalization of sum-product problem. The general principle of it was formulated in the Erdos-Szemeredi’s hypothesis. Instead of the Minkowski sum in this hypothesis, the set of values f(x,y) of a homogeneous polynomial f lin two variables, where x and y belong to subgroup G of is considered. The lower bound on the cardinality of such set is obtained. This topic has an applied value in the theory of information and dynamics in calculating the probabilities of events, as well as in various methods of encoding and decoding information.

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 incidences. Our extension is significantly more involved than the original proof, and in some sense runs the original technique a few times in a bootstrapping manner. We also study point–line incidences in the dual plane and in the double plane, developing analogs of the Szemerédi–Trotter theorem. As in the case of the sum–product problem, it turns out that the dual and double variants behave differently than the complex and real ones.

New Expander Bounds from Affine Group Energy

The purpose of this article is to further explore how the structure of the affine group can be used to deduce new incidence theorems, and to explore sum-product type applications of these incidence bounds, building on the recent work of Rudnev and Shkredov (On growth rate in $$SL_{2}(\mathbb {F}_p)$$ , the affine group and sum-product type implications, 2018. arXiv:1812.01671 ). We bound the energy of several systems of lines, in some cases obtaining a better energy bound than the corresponding bounds in Rudnev and Shkredov by exploiting a connection with collinear quadruples. Our motivation for seeking to generalise and improve the incidence bound from Rudnev and Shkredov comes from possible applications to sum-product problems. For example, we prove that, for any finite $$A \subset {\mathbb {R}}$$ the following superquadratic bound holds: This improves on a bound with exponent 2 that was given in Rudnev and Shkredov. We also give a threshold-beating asymmetric sum-product estimate for sets with small sum set by proving that there exists a positive constant c such that for all finite $$A,B \subset {\mathbb {R}}$$ ,

Distance and sum–product problems over finite p‐adic rings

Distance and sum–product problems over finite p‐adic rings

Conditional expanding bounds for two-variable functions over arbitrary fields

In this paper, we prove some results on the sum-product problem over arbitrary fields which improve and generalize results given by Hegyvári and Hennecart [5]. More precisely, we prove that, for related pairs of two-variable functions f(x,y) and g(x,y), if A and B are two sets in an arbitrary field F with |A|=|B|, thenmax⁡{|f(A,B)|,|g(A,B)|}≫|A|1+c, for some c>0.

Expansion for cubes in the Heisenberg group

Abstract In the last decade the growth in groups focused the interest of many researchers in the domain of additive combinatorics. In this note, we investigate the special case of the Heisenberg group ℋ {\mathcal{H}} of order 3 on different fields R, namely, R = 𝐑 {R=\mathbf{R}} and R = 𝐅 p {R=\mathbf{F}_{p}} . Denoting by [ x , y , z ] {[x,y,z]} the elements of ℋ {\mathcal{H}} , a cube is a subset of the type [ A , B , C ] {[A,B,C]} , with A , B , C ⊂ R {A,B,C\subset R} . This study relies to the well-known sum-product problem and its extensions, from which we obtain strong lower bounds for the size of the square [ A , B , C ] 2 {[A,B,C]^{2}} of cubes in ℋ {\mathcal{H}} . In the final section, we also consider the problem of counting the subsets of ℋ {\mathcal{H}} of the type [ A , B , C ] 2 {[A,B,C]^{2}} .

Variations on the Sum-Product Problem II

This paper is a sequel to a paper entitled Variations on the sum-product problem by the same authors [SIAM J. Discrete Math., 29 (2015), pp. 514-540]. In this sequel, we quantitatively improve several of the main results of the first paper as well as generalize a method from it to give a near-optimal bound for a new expander. The main new results are the following bounds, which hold for any finite set $A \subset \mathbb R$: $\exists a \in A$ such that $|A(A+a)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{186}}, |A(A-A)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{34}}, |A(A+A)| \gtrsim |A|^{\frac{3}{2}+\frac{5}{242}}, |\{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i \in A \}| \gg \frac{|A|^2}{\log |A|}$.

Conditional expanding bounds for two-variable functions over finite valuation rings

In this paper, we use methods from spectral graph theory to obtain some results on the sum–product problem over finite valuation rings R of order qr which generalize recent results given by Hegyvári and Hennecart (2013). More precisely, we prove that, for related pairs of two-variable functions f(x,y) and g(x,y), if A and B are two sets in R∗ with |A|=|B|=qα, then max{|f(A,B)|,|g(A,B)|}≫|A|1+Δ(α), for some Δ(α)>0.

If (A+A)/(A+A) is small, then the ratio set is large

In this paper, we consider the sum-product problem of obtaining lower bounds for the size of the set $$\frac{A+A}{A+A}:=\left \{ \frac{a+b}{c+d} : a,b,c,d \in A, c+d \neq 0 \right\},$$ for an arbitrary finite set $A$ of real numbers. The main result is the bound $$\left| \frac{A+A}{A+A} \right| \gg \frac{|A|^{2+\frac{2}{25}}}{|A:A|^{\frac{1}{25}}\log |A|},$$ where $A:A$ denotes the ratio set of $A$. This improves on a result of Balog and the author (arXiv:1402.5775), provided that the size of the ratio set is subquadratic in $|A|$. That is, we establish that the inequality $$\left| \frac{A+A}{A+A} \right| \ll |A|^{2} \Rightarrow |A:A| \gg \frac{ |A|^2}{\log^{25}|A|} . $$ This extremal result answers a question similar to some conjectures in a recent paper of the author and Zhelezov (arXiv:1410.1156).

Polynomials vanishing on grids: The Elekes-Rónyai problem revisited

In this paper we characterize real bivariate polynomials which have a small range over large Cartesian products. We show that for every constant-degree bivariate real polynomial $f$, either $|f(A,B)|=\Omega(n^{4/3})$, for every pair of finite sets $A,B\subset{\Bbb R}$, with $|A|=|B|=n$ (where the constant of proportionality depends on $\deg f$), or else $f$ must be of one of the special forms $f(u,v)=h(\varphi(u)+\psi(v))$, or $f(u,v)=h(\varphi(u)\cdot\psi(v))$, for some univariate polynomials $\varphi,\psi,h$ over ${\Bbb R}$. This significantly improves a result of Elekes and R\'onyai~(2000). Our results are cast in a more general form, in which we give an upper bound for the number of zeros of $z=f(x,y)$ on a triple Cartesian product $A\times B\times C$, when the sizes $|A|$, $|B|$, $|C|$ need not be the same; the upper bound is $O(n^{11/6})$ when $|A|=|B|=|C|=n$, where the constant of proportionality depends on $\deg f$, unless $f$ has one of the aforementioned special forms. This result provides a unified tool for improving bounds in various Erd\H os-type problems in geometry and additive combinatorics. Several applications of our results to problems of these kinds are presented. For example, we show that the number of distinct distances between $n$ points lying on a constant-degree algebraic curve that has a polynomial parameterization, and that does not contain a line, in any dimension, is $\Omega(n^{4/3})$, extending the result of Pach and de Zeeuw~(2014) and improving the bound of Charalambides~(2014), for the special case where the curve under consideration has a polynomial parameterization. We also derive improved lower bounds for several variants of the sum-product problem in additive combinatorics.

Difference sets are not multiplicatively closed

Difference sets are not multiplicatively closed, Discrete Analysis 2016:17, 20pp. The famous sum-product problem of Erdős and Szemeredi asks the following. Let $A$ be a set of $n$ real numbers. Define the _sumset_ $A+A$ of $A$ to be the set $\{x+y:x,y\in A\}$ and the _product set_ $A.A$ to be the set $\{xy:x,y\in A\}$. Must at least one of $A+A$ and $A.A$ have size $n^{2-o(1)}$? Since neither set can have size greater than $n^2$, this is saying that the only way of making one of them a little smaller is to make the other one have almost maximal size. In particular, arithmetic progressions have very large product sets and geometric progressions have very large sumsets. Until recently the best known bound was $n^{4/3}$, up to logarithmic factors [3], though the exponent has now been very slightly improved [2]. More generally, the _sum-product phenomenon_ is a collection of statements that say that in various senses it is not possible for a set to be additively structured and multiplicatively structured at the same time. For example, an important result of Bourgain, Katz and Tao states that if $A$ is a subset of a finite field ${\mathbb F}_p$ and $p^{\delta} < |A| < p^{1-\delta}$, then one of $A+A$ and $A.A$ must have size at least $n^{1+\epsilon}$ for some $\epsilon$ that depends on $\delta$ only. An extra difficulty here is that finite fields can have nontrivial subfields, so somehow the proof has to use the fact that $\mathbb F_p$ does not. This paper gives an interesting twist on the sum-product phenomenon. The underlying philosophy is still that a set with additive structure cannot be too multiplicatively structured at the same time. However, the notion of additive structure is different. Instead of assuming that the set has a small sumset, Shkredov assumes that it is a difference set of another set. This is a genuinely different assumption, since if $A$ has a large difference set $D=A-A$, it is perfectly possible for $D+D$ to be much bigger than $D$. Nevertheless, when $A$ is a set of real numbers, it still implies, as the main result of the paper shows, that the product set $D.D$ has size at least $|D|^{1+c}$, where $c>0$ is an absolute constant. The same is true of the quotient set $D/D$. The paper also proves a result of a similar kind in the finite field $\mathbb F_p$ provided that $D$ is not too big. Since multiplicative subgroups have, by definition, very small product sets, this result implies in particular that a multiplicative subgroup $H$ of $\mathbb F_p$ that is not too large is not the difference set of any subset of $\mathbb F_p$. (The precise assumption required is that $H$ has order at most $p^{4/5-\epsilon}$.) [1] Jean Bourgain, Nets Katz and Terence Tao, _A sum-product estimate in finite fields, and applications_, GAFA, 14 (2004), 27-57, also available at [arxiv:0301343](https://arxiv.org/abs/math/0301343) [2] Sergei Konyagin and Ilya D, Shkredov, _On sum sets of sets, having small product set_, [arxiv:1503.05771](https://arxiv.org/abs/1503.05771) [3] Jozsef Solymosi, Bounding multiplicative energy by the sumset , Adv. in Math, 222 (2009), 402-408, also available at [arxiv:0806.1040](https://arxiv.org/abs/0806.1040) [Article image by Lee Coursey](https://www.flickr.com/photos/leeco/14514643440/in/photolist-o7BkP7-wbsvhS-87Eun-hFM6Pi-5wWnAE-ixoVt1-3uoZCv-rujdcL-nDrm5X-nASM8G-rrtCFn-7rZcJ-ciosiU-7g3EA-bmx166-hVkB6H-ptbTJY-7oA7X5-aXrfFV-bZ8qg1-mhLdxq-fDgqzF-7PNJ2e-ee44aR-efNP5W-rxKAnt-bW3rkY-4i25j5-eYtYcc-52bQM8-h7NLYq-eyVP1U-gipwap-4zibnz-ts19q8-7Sfrp9-qZ9Tgj-7q9giY-qxrbPZ-GJXD6y-5RT9F9-aAgwaC-dVCMcL-6y8utz-4uNzTu-a7WQPt-4RoPSW-8yyZHX-5Bcypu-o3vgdb)

New Sum-Product Estimates for Real and Complex Numbers

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite set $$A$$A of positive real numbers, it is true that $$\begin{aligned} \Big |\Big \{\frac{a+b}{c+d}:a,b,c,d\in {A}\Big \}\Big |\ge {2|A|^2-1}. \end{aligned}$$|{a+bc+d:a,b,c,d?A}|?2|A|2-1.As a consequence of this result, it is also established that $$\begin{aligned} |4^{k-1}A^{(k)}|:=|\underbrace{\underbrace{A\ldots {A}}_{k\,\,\text {times}}+\cdots {+A\ldots {A}}}_{4^{k-1} \mathrm{times}}|\ge {|A|^k}. \end{aligned}$$|4k-1A(k)|:=|A?A?ktimes+?+A?A?4k-1times|?|A|k.Later on, it is shown that both of these bounds hold in the case when $$A$$A is a finite set of complex numbers, although with smaller multiplicative constants.