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58 votes
3 answers
6k views

Number of elements in the set $\{1,\cdots,n\}\cdot\{1,\cdots,n\}$

Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?
Kamalakshya's user avatar
7 votes
1 answer
760 views

Difference Sets

Suppose $$ P \subseteq \{1,2,\dots,N\},\quad |P| = K $$ We calculate the differences as: $$d=p_i-p_j\mod N,\quad i\ne j$$ Now let $a_d$ denote the number of occurrence of $d$ (for $d = 1, 2, \dots , N ...
Mahdi Khosravi's user avatar
35 votes
5 answers
4k views

Cliques, Paley graphs and quadratic residues

A question I've thought about, on and off for a long time, is how to improve the best bounds that (seem to be) known for the clique numbers of Paley graphs. If p=1 mod 4 is a prime, we can define the ...
Mike's user avatar
  • 703
18 votes
1 answer
890 views

Two conjectures about zero inner products and dissociated sets

The following problems come from something I worked on (with my coauthors) related to proving a new time lower bound for streaming problems. Having worked on these problems for some time with little ...
Simd's user avatar
  • 3,377
19 votes
3 answers
1k views

The sum of integers being a bijection

What are the pairs $(P,Q)$ of subsets of $\mathbb N$ for which the map \begin{eqnarray*} P\times Q & \rightarrow & {\mathbb N} \\\\ (p,q) & \mapsto & p+q \end{eqnarray*} is a bijection ...
Denis Serre's user avatar
  • 52.3k
3 votes
0 answers
193 views

A conjectural lower bound for $|\{\sum_{k=1}^nka_k:\ a_1,\ldots,a_n\ \text{are distinct elements of }\ A\}|$

Motivated by Question 315568 of mine, I'm interested in the set $$S(n):=\bigg\{\sum_{k=1}^n k\pi(k):\ \pi\in S_n\bigg\}.$$ It is easy to see that $$S(1)=\{1\},\ S(2)=\{4,5\}\ \text{and}\ S(3)=\{10,...
Zhi-Wei Sun's user avatar
  • 15.6k
113 votes
7 answers
8k views

Is the set $ AA+A $ always at least as large as $ A+A $?

Let $A$ be a finite set of real numbers. Is it always the case that $|AA+A| \geq |A+A|$? My first instinct is that this is obviously true, and there is a one-line proof which I am foolishly ...
Oliver Roche-Newton's user avatar
26 votes
2 answers
1k views

Partitions to different parts not exceeding $n$

Consider the polynomial $(1+x)(1+x^2)\dots (1+x^n)=1+x+\dots+x^{n(n+1)/2}$, which enumerates subj. How to prove that it's coefficients increase up to $x^{n(n+1)/4}$ (and hence decrease after this)? Or ...
Fedor Petrov's user avatar
23 votes
3 answers
3k views

How many different numbers can be obtained as product of first $n$ natural numbers?

Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set $\{1,2,\ldots,n\}$, that is consider the set $\{1^{a_1} \cdot 2^{...
Hujdurovic's user avatar
18 votes
4 answers
2k views

Number of vectors so that no two subset sums are equal

Consider all $10$-tuple vectors each element of which is either $1$ or $0$. It is very easy to select a set $v_1,\dots,v_{10}= S$ of $10$ such vectors so that no two distinct subsets of vectors $S_1 \...
Simd's user avatar
  • 3,377
5 votes
2 answers
516 views

Anticoncentration of the convolution of two characteristic functions

Edit: This is a question related to my other post, stated in a much more concrete way I think. I am interested in anything (ideas, references) related to the following problem: Suppose that $A \...
Maciej Skorski's user avatar
2 votes
1 answer
179 views

Only trivial solution to a pair of constrained linear diophantine equations

Given positive integer $n$, we are looking for a set of $n$ positive integers $a_i$. The following linear integer program must have only the trivial integer solution of all ones. $0 \le x_i \le \frac{...
joro's user avatar
  • 25.4k
1 vote
1 answer
195 views

Covering a finite ring with arithmetic progressions

Let $n\geq 2$ be a positive integer and $k$ be a number between $1$ and $n$. Recently, I came across the following question about $\mathbb Z/n\mathbb Z$ and I wonder if it was studied before. I'd be ...
Anton's user avatar
  • 1,625
59 votes
2 answers
4k views

For a finite set A of positive reals, prove that the set A + A - A contains at least as many positive as negative elements

I am currently working on a proof that would need to use the following theorem that I cannot prove: "Let $A$ be a finite set of positive real numbers. Then, the set $A + A - A$ contains at least ...
Timo Reichert's user avatar
28 votes
3 answers
969 views

Ordering subsets of the cyclic group to give distinct partial sums

Suppose that you are given a set $S$ of $k$ nonzero elements from $\mathbb{Z}_n$. Is it always possible to order the elements of $S$, say $a_1,a_2,\dots,a_k$ in such a way that the partial sums $a_1,...
Ian Wanless's user avatar
26 votes
0 answers
910 views

Which sets of roots of unity give a polynomial with nonnegative coefficients?

The question in brief:   When does a subset $S$ of the complex $n$th roots of unity have the property that $$\prod_{\alpha\, \in \,S} (z-\alpha)$$ gives a polynomial in $\mathbb R[z]$ with ...
Louis Deaett's user avatar
  • 1,513
17 votes
1 answer
1k views

Sum and product estimate over integers, rationals, and reals

My question is the following: is finding a lower bound for $|A+A\cdot A|$ (as a function of $|A|$) where $A$ is any finite subset of the positive integers equivalent to finding the same lower bound ...
George Shakan's user avatar
17 votes
2 answers
2k views

A proof of Van der Waerden's theorem using a weakened form of Szemeredi's theorem

Van der Waerden's theorem states that any colouring of the integers in a finite number of colours has monochromatic arithmetic progressions of arbitrary length. Szemerédi's Theorem is a dramatic ...
Ivan Meir's user avatar
  • 4,862
15 votes
1 answer
417 views

What is the smallest cardinality of a self-linked set in a finite cyclic group?

A subset $A$ of a group $G$ is defined to be self-linked if $A\cap gA\ne\emptyset$ for all $g\in G$. This happens if and only if $AA^{-1}=G$. For a finite group $G$ denote by $sl(G)$ the smallest ...
Taras Banakh's user avatar
  • 41.8k
14 votes
1 answer
633 views

Minimal "sumset basis" in the discrete linear space $\mathbb F_2^n$

For a set $C\subseteq \mathbb F_2^n$, let $2C=C+C:=\{\alpha+\beta\colon \alpha,\beta\in C\}$. I want to find $C$ of the smallest possible size such that $2C=\mathbb F_2^n$. Let $m(n)$ be the size of a ...
pointer's user avatar
  • 197
13 votes
1 answer
468 views

Near-linear mappings from $\mathbb F_p$ to $\mathbb R$

$\newcommand{\F}{{\mathbb F}}$ $\newcommand{\R}{{\mathbb R}}$ $\renewcommand{\phi}{\varphi}$ Let $p\ge 5$ be a prime. If the functions $\phi_1,\phi_2,\phi_3\colon\F_p\to\R$ satisfy $\phi_1(x)+\...
Seva's user avatar
  • 23k
11 votes
2 answers
410 views

Extension of Dickson's theorem on integers of the form $a^2+b^2+2c^2$

Theorems V in this paper of L.E. Dickson states that the following two sets are equal. $$E=\{a^2+b^2+2c^2 \ | \ a,b,c \in \mathbb{Z}\} \ \text{ and } \ F=\mathbb{N} \setminus \{4^k(16n+14) \ | \ k,n \...
Sebastien Palcoux's user avatar
10 votes
4 answers
1k views

Binomial coefficient in Andrews' partition book

First of all, I think MathOverflow is a very great community to discuss math, either basic or advanced, and I'm glad to participate here. It's my first post, so I'm sorry if i did anything wrong, and ...
Guilherme's user avatar
  • 103
9 votes
0 answers
297 views

An abstract zero-sum problem

I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...
monkeymaths's user avatar
  • 1,169
8 votes
1 answer
380 views

Question about estimating random symmetric sums modulo p

Let $n > 0$ be a positive integer (large) and $p > 2$ a fixed prime number. What is the probability that $$\sum_{ 1 \leq i < j \leq n} a_ia_j = 0 \mod p$$ where $a_1, a_2, \dots a_n$ are ...
shurtados's user avatar
  • 1,101
8 votes
0 answers
304 views

A strong sum-product "for translates" in finite fields

In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting? Proposition There exists an absolute constant $c$ such ...
Nick Gill's user avatar
  • 11.2k
7 votes
2 answers
845 views

Decomposition of a natural number as sum of positive integers

Let $n \in \mathbb{N}$ be a positive natural number and denote by $f(n)$ the number of decompositions of $n$ of the form $n = a+b+c+d$ where $a,b,c,d > 0$ are also positive natural numbers such ...
Puzzled's user avatar
  • 8,998
7 votes
1 answer
569 views

Upper bound for size of subsets of a finite group that contains a sum-full set

Problem I'm looking for an upper bound for the number $k(G)$ of a finite group $G$, defined as follow: Let $\mathcal{F}_k$ be the family of subsets of $G$ with size $k$, and we define $k(G)$ be ...
Hsien-Chih Chang 張顯之's user avatar
7 votes
1 answer
399 views

Maximum size of $k$-Sidon set over $\mathbb{F}_2^n.$

Fix $k \in \mathbb{N}$, $k \ge 2.$ Does there exist a subset $A \subset \mathbb{F}_2^n$ such that $|A| \ge c 2^{n/k}$ with some absolutely positive constant $c,$ and satisfying $$ a_1 + a_2 + \...
user avatar
7 votes
1 answer
509 views

A permutation problem

Here I ask a question on permutations of $n$ distinct real numbers. QUESTION: Let $a_1,a_2,\ldots,a_n\ (n>1)$ be (pairwise) distinct real numbers. Is there a permutation $b_1,\ldots,b_n$ of $a_1,\...
Zhi-Wei Sun's user avatar
  • 15.6k
5 votes
1 answer
261 views

Divisibility labeling on a boolean lattice and nonzero Euler totient

Let $B_n$ be the subset lattice of $\{1,2, \dots , n \}$, also called the boolean lattice of rank $n$. A labeling $f: B_n \to \mathbb{N}_{\ge 1}$ is called acceptable if $\forall a,b \in B_n$: ...
Sebastien Palcoux's user avatar
4 votes
2 answers
427 views

How big must the sumset $A+A$ be if $A$ satisfies no translation-invariant equations of low height?

Suppose $A$ is a finite subset of an abelian group. If there is no solution to $ma+nb=(m+n)c$ with $0\leq m,n\leq M$, can we bound $|A+A|$ from below? I am interested if one can obtain bounds much ...
deadcat's user avatar
  • 41
4 votes
1 answer
373 views

Difference set of difference set

I am a hobby computer scientist and searching for an algorithm to construct a set of n numbers (integers) with certain properties. Property 1 / Step 1 All pairwise differences of the elements should ...
BenBar's user avatar
  • 73
3 votes
0 answers
144 views

Flat polynomials with factors of big height

Let $p(x)$ be a polynomial of degree $n$ with all coefficients in $\{-1,0,1\}$ (such polynomials are sometimes called flat). I am wondering how big the coefficients of a factor of $p$ can be. Call ...
Wolfgang's user avatar
  • 13.4k
3 votes
0 answers
187 views

Szemerédi’s theorem in really dense sets

This question is inspired by Tao’s answer in this post. I have thought about this occasionally for several months without anything concrete. Question: Given $\delta>0$ and $k\ge 3$, let $N= N_k(\...
Zach Hunter's user avatar
  • 3,499
3 votes
1 answer
318 views

Problem related to Frobenius coin problem

Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$. Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ good if, for any $r,s,u,...
Turbo's user avatar
  • 13.9k
3 votes
3 answers
748 views

Is the sumset or the sumset of the square set always large?

Let A be a finite subset of $\mathbb{N}$, $\mathbb{R}$, or a sufficiently small subset of $\mathbb{F}_{p}$. Do we have a lower bound of the form $|A|^{1+\delta}$ on the following quantity: $$\max (|\...
Mark Lewko's user avatar
3 votes
1 answer
260 views

Davenport constant $D(S_5)=10$ or $11$?

I am working on computing the Davenport constant $D(G)$ of symmetric groups, which is the minimal number $d$ such that every sequence of $d$ elements, possibly with repetitions, is one-product, i.e. ...
Mikel Martinez Puente's user avatar
2 votes
0 answers
143 views

A set in Z/nZ which contains two elements, one of which is a small multiple of the other

While doing my research, I came across yet another problem in $\mathbb Z/n\mathbb Z$ (see my previous question on a related matter here). Let $n$ be a prime and let $k$ be an integer, $1 \leq k \leq ...
Anton's user avatar
  • 1,625
2 votes
1 answer
621 views

On 'Improved Bounds for the Sunflower Lemma' [Alweiss, Lovett, Wu, Zhang]

I have been reading the paper 'Improved Bounds for the Sunflower Lemma' (Ann. of Math., Vol. 194(3), pp. 795-815), and have not managed to understand the following: I would like a formalization for ...
user avatar
1 vote
0 answers
147 views

Counting Hamiltonian cycles in graph and finding a coefficient of polynomial

Exact result is #P-Hard, so we are looking for bounds. Let $G$ be simple graph or simple digraph and $A$ its adjacency matrix. $A$ is $n \times n$ with entries only zeros or ones. Let $K=\mathbb{Z}[...
joro's user avatar
  • 25.4k
1 vote
1 answer
466 views

Some divisibility constraints in Frobenius coin problem

Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$. Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ excellent if linear form ...
Turbo's user avatar
  • 13.9k