We added a conditional theorem to the paper, which holds under a certain generalization of Fermat's Last Theorem; and, we added an acknowledgment (to Jozsef Solymosi)
We show that under the assumption of a 24-term version of Fermat's Last Theorem, there exists an absolute constant c > 0 such that if S is a set of n > n_0 positive integers satisfying |S.S| < n^(1+c), then the sumset S.S satisfies |S+S| >> n^2. In other words, we prove a weak form of the Erdos-Szemeredi sum-product conjecture, conditional on an extension of Fermat's Last Theorem. Unconditionally, we prove this theorem for when S is a set of n monic polynomials. We also prove an analogue of a theorem of Bourgain and Chang for the ring C[x].
In this paper, we consider sums of generalized polygonal numbers with repeats, generalizing Fermat's polygonal number theorem which was proven by Cauchy. In particular, we obtain the minimal number of generalized $m$-gonal numbers required to represent every positive integer and we furthermore generalize this result to obtain optimal bounds when many of the generalized $m$-gonal numbers are repeated $r$ times, where $r\in\mathbb{N}$ is fixed.
We prove that a certain family of sums of products of three binomials has alternating behavior modulo a prime $p$. To accomplish this we rewrite these sums as signed sums of products of three binomials, the better to handle $p$, and we give closed-form expressions for two related sums of signed products of three binomials.
In this paper, we establish several formulae for sums and alternating sums of products of generalized Fibonacci and Lucas numbers. In particular, we recover and extend all results of Z. Cerin and Z. Cerin & G. M. Gianella, more easily.
In their seminal paper Erdős and Szemerédi formulated conjectures on the size of sumset and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph. In this paper we show that this strong form of the Erdős-Szemerédi conjecture does not hold. We give upper and lower bounds on the cardinalities of sumsets, product sets and ratio sets along the edges of graphs.
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.
Let $A \subset \mathbb R$ and $G \subset A \times A$. We prove that, for any $λ\in \mathbb R \setminus \{-1,0,1\}$, \[ \max \{|A+_G A|, |A+_G λA|, |A\cdot_G A|\} \gg |G|^{6/11}. \]
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 $A$.
Improving upon a technique of Croot and Hart, we show that for every $h$, there exists an $ε> 0$ such that if $A \subseteq \mathbb{R}$ is sufficiently large and $|A.A| \le |A|^{1+ε}$, then $|hA| \ge |A|^{Ω(e^{\sqrt{c\log{h}}})}$.
Let $A \subset \mathbb{F}_p$ of size at most $p^{3/5}$. We show $$|A+A| + |AA| \gtrsim |A|^{6/5 + c},$$ for $c = 4/305$. Our main tools are the cartesian product point--line incidence theorem of Stevens and de Zeeuw and the theory of higher energies developed by the second author.
In this paper we initiate the study of products and sums divisible by central binomial coefficients. We show that 2(2n+1)binom(2n,n)| binom(6n,3n)binom(3n,n) for every n=1,2,3,... Also, for any nonnegative integers $k$ and $n$ we have $$\binom {2k}k | \binom{4n+2k+2}{2n+k+1}\binom{2n+k+1}{2k}\binom{2n-k+1}n$$ and $$\binom{2k}k | (2n+1)\binom{2n}nC_{n+k}\binom{n+k+1}{2k},$$ where $C_m$ denotes the Catalan number $\binom{2m}m/(m+1)=\binom{2m}m-\binom{2m}{m+1}$. Applying this result we obtain two sums divisible by central binomial coefficients.
Suppose that $\mathbb{F}_p$ is coloured with $r$ colours. Then there is some colour class containing at least $c_r p^2$ quadruples of the form $(x, y , x + y, xy)$.
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 …
We prove that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1}=n_1$, and non-negative integers $j$ and $r$ with $j\leqslant m$, the following two expressions \begin{align*} &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack n_1}^{-1}\sum_{k=0}^{n_1} q^{j(k^2+k)-(2r+1)k}[2k+1]^{2r+1}\prod_{i=1}^m {n_i+n_{i+1}+1\brack n_i-k},\\[5pt] &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack n_1}^{-1}\sum_{k=0}^{n_1}(-1)^k q^{{k\choose 2}+j(k^2+k)-2rk}[2k+1]^{2r+1}\prod_{i=1}^m {n_i+n_{i+1}+1\brack n_i-k} \end{align*} are Laurent polynomials in $q$ with integer coefficients, where $[n]=1+q+\cdots+q^{n-1}$ and ${n\brack k…
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…
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}.
We formulate conjectures regarding the maximum value and maximizing matrices of the permanent and of diagonal products on the set of stochastic matrices with bounded rank. We formulate equivalent conjectures on upper bounds for these functions for nonnegative matrices based on their rank, row sums and column sums. In particular we conjecture that the permanent of a singular nonnegative matrix is bounded by 1/2 times the minimum of the product of its row sums and the product of its column sums, and that the product of the elements of any diagonal of a singular nonnegative matrix is bounded by 1…
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 $10AB = \F_q.$
Our aim in this paper is to show that, for any $k$, there is a finite colouring of the set of rationals whose denominators contain only the first $k$ primes such that no infinite set has all of its finite sums and products monochromatic. We actually prove a `uniform' form of this: there is a finite colouring of the rationals with the property that no infinite set whose denominators contain only finitely many primes has all of its finite sums and products monochromatic. We also give various other results, including a new short proof of the old result that there is a finite colouring of the natu…
In this paper we show that for any $k\geq2$, there exist two universal constants $C_k,D_k>0$, such that for any finite subset $A$ of positive real numbers with $|AA|\leq M|A|$, $|kA|\geq \frac{C_k}{M^{D_k}}\cdot|A|^{\log_42k}.$
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.
Hindman proved that, whenever the set $\mathbb{N}$ of naturals is finitely colored, there must exist non-constant monochromatic solution of the equation $a+b=cd$. In this paper we extend this result for dense subsemigroups of $((0, \infty), +)$ to near zero.
In this article, we consider a proper total coloring distinguishes adjacent vertices by sums, if every two adjacent vertices have different total sum of colors of the edges incident to the vertex and the color of the vertex. Pilsniak and Wozniak \cite{PW} first introduced this coloring and made a conjecture that the minimal number of colors need to have a proper total coloring distinguishes adjacent vertices by sums is less than or equal to the maximum degree plus $3$. We study proper total colorings distinguishing adjacent vertices by sums of some graphs and their products. We find that these…
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.
Let $Θ(G)$ denote the Shannon capacity of a graph $G$. We give an elementary proof of the equivalence, for any graphs $G$ and $H$, of the inequalities $Θ(G\sqcup H)>Θ(G)+Θ(H)$ and $Θ(G\boxtimes H)>Θ(G)Θ(H)$. This was shown independently by Wigderson and Zuiddam [2022] using Kadison-Dubois duality and the Axiom of choice.
A set in d dimensional Euclidean space with d larger than 2 having Hausdorff dimension at least d/2 must have distance set with Hausdorff dimension strictly greater than 1/2.
It was proved that whenever N is partitioned into finitely many cells, one cell must contain arbitrary length geo-arithmetic progressions. It was also proved that arithmetic and geometric progressions can be nicely intertwined in one cell of partition, whenever N is partitioned into finitely many cells. In this article we shall prove that similar types of results also hold near zero in some suitable dense subsemigroup S of ((0; 1) ; +), using the Stone- Cech compactification of ?S.
We show that if the moment-angle complex $\mathcal{Z}_K$ associated to a simplicial complex $K$ is homotopy equivalent to a connected sum of sphere products with two spheres in each product, then $K$ decomposes as the simplicial join of an $n$-simplex $Δ^n$ and a minimally non-Golod complex. In particular, we prove that $K$ is minimally non-Golod for every moment-angle complex $\mathcal{Z}_K$ homeomorphic to a connected sum of two-fold products of spheres, answering a question of Grbić, Panov, Theriault and Wu.
We show that, for Hankel matrices, total nonnegativity (resp. total positivity) of order r is preserved by sum, Hadamard product, and Hadamard power with real exponent t \ge r-2. We give examples to show that our results are sharp relative to matrix size and structure (general, symmetric or Hankel). Some of these examples also resolve the Hadamard critical-exponent problem for totally positive and totally nonnegative matrices.
