An improved bound on the sum-product estimate in $\mathbb{F}_{p}$
Connor Paul Wilson
备注：
The note was submitted as part of a larger portfolio, which is where the term "improved" comes from, however, this has caused confusion as it seems to present itself as improving the current standing of the problem as a whole
We give an improved bound on the famed sum-product estimate in a field of residue class modulo $p$ ($\mathbb{F}_{p}$) by Erdős and Szemeredi, and a non-empty set $A \subset \mathbb{F}_{p}$ such that: $$ \max \{|A+A|,|A A|\} \gg \min \left\{\frac{|A|^{15 / 14} \max \left\{1,|A|^{1 / 7} p^{-1 / 14}\right\}}{(\log |A|)^{2 / 7}}, \frac{|A|^{11 / 12} p^{1 / 12}}{(\log |A|)^{1 / 3}}\right\}, $$ and more importantly: $$\max \{|A+A|,|A A|\} \gg \frac{|A|^{15 / 14}}{(\log |A|)^{2 / 7}}.$$
A zero-sum sequence over ${\mathbb Z}$ is a sequence with terms in ${\mathbb Z}$ that sum to $0$. It is called minimal if it does not contain a proper zero-sum subsequence. Consider a minimal zero-sum sequence over ${\mathbb Z}$ with positive terms $a_1,\ldots,a_h$ and negative terms $b_1,\ldots,b_k$. We prove that $h\leq \lfloor σ^+/k\rfloor$ and $k\leq \lfloor σ^+/h\rfloor$, where $σ^+=\sum_{i=1}^h a_i=-\sum_{j=1}^k b_j$. These bounds are tight and improve upon previous results. We also show a natural partial order structure on the collection of all minimal zero-sum sequences over the set $\…
A set $A\subset \mathbb{F}_p^n$ is sum-free if $A+A$ does not intersect $A$. If $p\equiv 2 \mod 3$, the maximal size of a sum-free in $\mathbb{F}_p^n$ is known to be $(p^n+p^{n-1})/3$. We show that if a sum-free set $A\subset \mathbb{F}_p^n$ has size at least $p^n/3-p^{n-1}/6+p^{n-2}$, then there exists subspace $V<\mathbb{F}_p^n$ of co-dimension 1 such that $A$ is contained in $(p+1)/3$ cosets of $V$. For $p=5$ specifically, we show the stronger result that every sum-free set of size larger than $1.2\cdot 5^{n-1}$ has this property, thus improving on a recent theorem of Lev.
In this paper we prove that every subset of $\mathbb{F}_p^2$ meeting all $p+1$ lines passing through the origin has a zero-sum subset. This is motivated by a result of Gao, Ruzsa and Thangadurai which states that $OL(\mathbb{F}_{p}^{2})=p+OL(\mathbb{F}_{p})-1$, for sufficiently large primes $p$. Here $OL(G)$ denotes the so-called Olson constant of the additive group $G$ and represents the smallest integer such that no subset of cardinality $OL(G)$ is zero-sum-free. Our proof is in the spirit of the Combinatorial Nullstellensatz.
In this paper we give a different approach to determining the cardinality of $h$-fold sumsets $hA$ when $A\subset \mathbb{Z}^d$ has $d+2$ elements. This enables us to provide more general result with a shorter and simpler proof. We also obtain an upper bound for the value of $|hA|$ when $A\subset \mathbb{Z}^d$ is a set of $d+3$ elements with simplicial hull.
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 …
This paper proves a bound for exponential sums over orbits of vectors in $\mathbb{F}_p^d$ under subgroups of $\rm{GL}_d(\mathbb{F}_p)$. The main tool is a classification theorem for approximate groups due to Gill, Helfgott, Pyber, and Szabó.
The $3k-4$ conjecture in groups $\mathbb{Z}/p\mathbb{Z}$ for $p$ prime states that if $A$ is a nonempty subset of $\mathbb{Z}/p\mathbb{Z}$ satisfying $2A\neq \mathbb{Z}/p\mathbb{Z}$ and $|2A|=2|A|+r \leq \min\{3|A|-4,\;p-r-4\}$, then $A$ is covered by an arithmetic progression of size at most $|A|+r+1$. A theorem of Serra and Zémor proves the conjecture provided $r\leq 0.0001|A|$, without any additional constraint on $|A|$. Subject to the mild additional constraint $|2A|\leq 3p/4$ (which is optimal in a sense explained in the paper), our first main result improves the bound on $r$, allowing $r…
Let $A$ be a set in an abelian group $G$. For integers $h,r \geq 1$ the generalized $h$-fold sumset, denoted by $h^{(r)}A$, is the set of sums of $h$ elements of $A$, where each element appears in the sum at most $r$ times. If $G=\mathbb{Z}$ lower bounds for $|h^{(r)}A|$ are known, as well as the structure of the sets of integers for which $|h^{(r)}A|$ is minimal. In this paper we generalize this result by giving a lower bound for $|h^{(r)}A|$ when $G=\mathbb{Z}/p\mathbb{Z}$ for a prime $p$, and show new proofs for the direct and inverse problems in $\mathbb{Z}$.
Let $\mathbb{F}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a nonempty subset of $\mathbb{F}_p$. In this paper we show that if $|A|\preceq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succeq|A|^{13/12};\] if $|A|\succeq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succapprox \min\{|A|^{13/12}(\frac{|A|}{p^{0.5}})^{1/12},|A|(\frac{p}{|A|})^{1/11}\}.\] These results slightly improve the estimates of Bourgain-Garaev and Shen. Sum-product estimates on different sets are also considered.
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.
Let $A$ be a multiplicative subgroup of $\mathbb Z_p^*$. Define the $k$-fold sumset of $A$ to be $kA=\{x_1+\dots+x_k:x_i \in A,1\leq i\leq k\}$. We show that $6A\supseteq \mathbb Z_p^*$ for $|A| > p^{\frac {11}{23} +ε}$. In addition, we extend a result of Shkredov to show that $|2A|\gg |A|^{\frac 85-ε}$ for $|A|\ll p^{\frac 59}$.
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}).
Let $A$ and $B$ be additive sets of $\mathbb{Z}_{2k}$, where $A$ has cardinality $k$ and $B=v.\complement A$ with $v\in\mathbb{Z}_{2k}^{\times}$. In this note some bounds for the cardinality of $A+B$ are obtained, using four different approaches. We also prove that in a special case the bound is not sharp and we can recover the whole group as a sumset.
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 $10AB = \F_q.$
Let $G = (V,E)$ be a finite graph and let $(\mathbb{A},+)$ be an abelian group with identity 0. Then $G$ is \textit{$\mathbb{A}$-magic} if and only if there exists a function $ϕ$ from $E$ into $\mathbb{A} - \{0\}$ such that for some $c \in \mathbb{A}$, $\sum_{e \in E(v)} ϕ(e) = c$ for every $v \in V$, where $E(v)$ is the set of edges incident to $v$. Additionally, $G$ is \textit{zero-sum $\mathbb{A}$-magic} if and only if $ϕ$ exists such that $c = 0$. We consider zero-sum $\mathbb{A}$-magic labelings of graphs, with particular attention given to $\mathbb{A} = \mathbb{Z}_{2j}^k$. For $j \geq 1$…
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.
Let $S$ be a set of $n\times n$ matrices over a field $\mathbb{F}$. We show that the $\mathbb{F}$-linear span of the words in $S$ of length at most $$2n\log_2n+4n$$ is the full $\mathbb{F}$-algebra generated by $S$. This improves on the $n^2/3+2/3$ bound by Paz (1984) and an $O\left(n^{1.5}\right)$ bound of Pappacena (1997).
The subspace sum graph $\mathcal{G}(\mathbb{V})$ on a finite dimensional vector space $\mathbb{V}$ was introduced by Das [Subspace Sum Graph of a Vector Space, arXiv:1702.08245], recently. The vertex set of $\mathcal{G}(\mathbb{V})$ consists of all the nontrivial proper subspaces of $\mathbb{V}$ and two distinct vertices $W_1$ and $W_2$ are adjacent if and only if $W_1+W_2=\mathbb{V}$. In that paper, some structural indices (e.g., diameter, girth, connectivity, domination number, clique number and chromatic number) were studied, but the characterization of automorphisms of $\mathcal{G}(\mathbb…
A subset $A$ of a finite abelian group is called $(k,\ell)$-sum-free if $kA \cap \ell A=\emptyset.$ In this paper, we extend this concept to compact abelian groups and study the question of how large a measurable $(k,\ell)$-sum-free set can be. For integers $1 \leq k <\ell$ and a compact abelian group $G$, let $$λ_{k,\ell}(G)=\sup\{ μ(A): kA \cap \ell A =\emptyset \}$$ be the maximum possible size of a $(k,\ell)$-sum-free subset of $G$. We prove that if $G=\mathbb{I} \times M$, where $\mathbb{I}$ is the identity component of $G$, then $$λ_{k, \ell}(G)=\max \left\{ λ_{k, \ell}(M), λ_{k, \ell}(\…
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.
In this paper, we give a lower bound for the maximum size of a $k$-colored sum-free set in $\mathbb{Z}_m^n$, where $k\geq 3$ and $m\geq 2$ are fixed and $n$ tends to infinity. If $m$ is a prime power, this lower bound matches (up to lower order terms) the previously known upper bound for the maximum size of a $k$-colored sum-free set in $\mathbb{Z}_m^n$. This generalizes a result of Kleinberg-Sawin-Speyer for the case $k=3$ and as part of our proof we also generalize a result by Pebody that was used in the work of Kleinberg-Sawin-Speyer. Both of these generalizations require several key new id…
Let $\mathbb{F}_{q}$ be a finite field with $q$ elements and $\mathbb{F}_{q}[x]$ the ring of polynomials over $\mathbb{F}_{q}$. Let $l(x), k(x)$ be coprime polynomials in $\mathbb{F}_{q}[x]$ and $Φ(k)$ the Euler function in $\mathbb{F}_{q}[x]$. Let $π(l, k; n)$ be the number of monic irreducible polynomials of degree $n$ in $\mathbb{F}_{q}[x]$ which are congruent to $l(x)$ module $k(x)$. For any positive integer $n$, we denote by $Ω(n)$ the least prime divisor of $n$. In this paper, we show that $$π(l, k; n)=\frac{1}{Φ(k)}\frac{q^{n}}{n}+O\left(n^α\right)+O\left(\frac{q^{\frac{n}{Ω{(n)}}}}{n}\…
In this paper, we discuss the adjacency matrices of finite undirected simple graphs over a finite prime field $\mathbb{F}_p$. We apply symmetric (row and column) elementary transformations to the adjacency matrix over $\mathbb{F}_p$ in order to get a direct sum decomposition by other adjacency matrices. In this paper, we give a complete description of the direct sum decomposition of the adjacency matrix of any graph over $\mathbb{F}_p$ for any odd prime $p$. Our key tool is quadratic residues of $\mathbb{F}_p$.
For an abelian group $G$ and an integer $t > 0$, the \emph{modified Erdös--Ginzburg--Ziv constant} $s_t'(G)$ is the smallest integer $\ell$ such that any zero-sum sequence of length at least $\ell$ with elements in $G$ contains a zero-sum subsequence (not necessarily consecutive) of length $t$. We compute $s_t'(G)$ for $G = \mathbb Z/n\mathbb Z$ and for $t = n$, $G = (\mathbb Z/n\mathbb Z)^2$.
A long standing question asks whether $\mathbb{Z}$ is uniformly 2-repetitive [Justin 1972, Pirillo and Varricchio, 1994], that is, whether there is an infinite sequence over a finite subset of $\mathbb{Z}$ avoiding two consecutive blocks of same size and same sum or not. Cassaigne \emph{et al.} [2014] showed that $\mathbb{Z}$ is not uniformly 3-repetitive. We show that $\mathbb{Z}^2$ is not uniformly 2-repetitive. Moreover, this problem is related to a question from Mäkelä in combinatorics on words and we answer to a weak version of it.
We give an improved upper bound for the Grünbaum--Hadwiger--Ramos problem: Let $d,n,k \in \mathbb{N}$ such that $d \geq 2^n(1+2^{k-1})$. Given $2^{n+1}$ masses on $\mathbb{R}^d$, there exist $k$ hyperplanes in $\mathbb{R}^d$ that partition it into $2^k$ sets of equal size with respect to all measures. This is an improvement to the previous bound $d \geq 2^{n + k}$ by Mani-Levitska, Vrećica & Živaljević in 2006. This is achieved by classifying the number of certain Gray code patterns modulo 2. The reduction was developed by Blagojević, Frick, Haase & Ziegler in 2016. It utilizes the group actio…
In this paper we introduce a graph structure, called subspace sum graph $\mathcal{G}(\mathbb{V})$ on a finite dimensional vector space $\mathbb{V}$ where the vertex set is the collection of non-trivial proper subspaces of a vector space and two vertices $W_1,W_2$ are adjacent if $W_1 + W_2=\mathbb{V}$. The diameter, girth, connectivity, maximal independent sets, different variants of domination number, clique number and chromatic number of $\mathcal{G}(\mathbb{V})$ are studied. It is shown that two subspace sum graphs are isomorphic if and only if the base vector spaces are isomorphic. Finally…
In this paper we present a procedure which allows to transform a subset $A$ of $\mathbb{Z}_{p}$ into a set $ A'$ such that $ |2\hspace{0.15cm}\widehat{} A'|\leq|2\hspace{0.15cm}\widehat{} A | $, where $2\hspace{0.15cm}\widehat{} A$ is defined to be the set $\left\{a+b:a\neq b,\;a,b\in A\right\}$. From this result, we get some lower bounds for $ |2\hspace{0.15cm}\widehat{} A| $. Finally, we give some remarks related to the problem for which sets $A\subset \mathbb{Z}_{p}$ we have the equality $|2\hspace{0.15cm}\widehat{} A|=2|A|-1$.
The Minkowski sum of two subsets $A$ and $B$ of a finite abelian group $G$ is defined as all pairwise sums of elements of $A$ and $B$: $A + B = \{ a + b : a \in A, b \in B \}$. The largest size of a $(k, \ell)$-sum-free set in $G$ has been of interest for many years and in the case $G = \mathbb{Z}/n\mathbb{Z}$ has recently been computed by Bajnok and Matzke. Motivated by sum-free sets of the torus, Kravitz introduces the noisy Minkowski sum of two sets, which can be thought of as discrete evaluations of these continuous sumsets. That is, given a noise set $C$, the noisy Minkowski sum is define…
An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair $\{x+y,xy\}$. We answer this question affirmatively in a strong sense by exhibiting a large new class of non-linear patterns which can be found in a single cell of any finite partition of $\mathbb{N}$. Our proof involves a correspondence principle which transfers the problem into the language of topological dynamics. As a corollary of our main theorem we obtain partition regularity for new types of equations, such as $x^2-y^2=z$ and $x^2+2y^2-3z^2=w$.
For a bivariate $P(x,y) \in \mathbb{R}[x,y]\setminus (\mathbb{R}[x] \cup \mathbb{R}[y])$, our first result shows that for all finite $A \subseteq \mathbb{R}$, $|P(A,A)|\geq α|A|^{5/4}$ with $α=α(\mathrm{deg} P) \in \mathbb{R}^{>0}$ unless $$ P(x,y)=f(γu(x)+δu(y)) \text{ or } P(x,y)=f(u^m(x)u^n(y)) $$ for some univariate $f, u \in \mathbb{R}[t]\setminus \mathbb{R}$, constants $γ, δ\in \mathbb{R}^{\neq 0}$, and $m, n\in \mathbb{N}^{\geq 1}$. This resolves the symmetric nonexpanders classification problem proposed by de Zeeuw. Our second and third results are sum-product type theorems for two pol…
