All Questions
122 questions
5
votes
0
answers
185
views
Gaps in sumsets and difference sets
a) Let $S\subset \{1,2,\dotsc,N\}$ be a fairly thick set (with at least $N^{1-\epsilon}$ elements, say). Suppose that the intersection of, say,
$$3 S - 3 S = \{a_1+a_2+a_3-(a_4+a_5+a_6): a_1,\dotsc,...
- 20.2k
7
votes
0
answers
176
views
Sumsets that contains many squares, Improvement on the bound
I'm being troubled by this problem on AoPS:
https://artofproblemsolving.com/community/c6h1998237p13955033
I searched for any literature related to it such as
Nguyen, Hoi H., and Van H. Vu., Squares ...
- 63
5
votes
0
answers
307
views
On $s$-additive sequences
For a non-negative integer $s$, a strictly increasing sequence of positive integers $\{a_n\}$ is called $s$-additive if for $n>2s$, $a_n$ is the least integer exceeding $a_{n-1}$ which has ...
- 791
3
votes
2
answers
224
views
Unique "clique" of differences in $\mathbb{Z}/m\mathbb{Z}$
Are there absolute constants $0 < \epsilon < 1$ and $N \in \mathbb{N}$ such that the following holds: For every $m \in \mathbb{N}$ and every $A \subseteq \mathbb{Z}/m\mathbb{Z}$ with $\frac{\...
- 73
2
votes
0
answers
278
views
On $(k,\ell)$-sumfree sets
Call a set $\mathcal S \subset \mathbb N$ to be $(k,\ell)$-sumfree if there are no non-trivial solutions to the equation
$$x_1+\dots +x_k = y_1+\dots +y_\ell$$
in the set (for distinct $x_i$'s and $...
- 791
5
votes
2
answers
691
views
Representing natural numbers as sums of distinct prime powers
I am investigating whether every natural number $n > 18$ can be represented as a sum $p_1^{m_1} + \dots + p_k^{m_k}$, where $p_1, \dots, p_k$ are distinct primes, and $m_1, \dots, m_k$ are distinct ...
- 1,228
16
votes
2
answers
1k
views
The Stable Set Conjecture
A set $\mathcal S$ of positive integers is called stable if for every fixed positive integer $d$, the relation
$$n\in \mathcal S \iff dn\in \mathcal S$$
holds for almost all positive integers $n$. ...
- 791
15
votes
2
answers
750
views
Subsets of $(\mathbb{Z}/p)^{\times n}$
There seems to be some combinatorial fact that every subset $A$ of $G=(\mathbb{Z}/p)^{\times n}$ of cardinality $\frac{p^n-1}{p-1}+1$ containing $\vec{0}$ satisfies $(p-1)A=G$. ($p$ is a prime number....
- 153
1
vote
1
answer
200
views
Instance of polynomial van der Waerden without good bounds
Let $P\subset \Bbb{Z}[X]$ be a finite set of polynomials with constant-term zero. Then, polynomial vdW says:
For eacg finite $r$, there exists some $N=N(P,r)$, such that every $r$-coloring $C:\{1,\...
- 3,499
3
votes
0
answers
140
views
Counting $A+A-A$ with partial multiplicity
A recent question asked whether, given a finite set of positive numbers $A$, it is always the case that the set $A+A-A$ has more positive than negative elements. Terry Tao showed that this is false (...
- 6,001
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 ...
- 643
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(\...
- 3,499
1
vote
0
answers
170
views
A representation problem involving strict partition numbers
For each positive integer $n$, let $q(n)$ denote the number of ways to write $n$ as a sum of distinct positive integers. We call those $q(n)\ (n=1,2,3,\ldots)$ strict partition numbers.
The sequence $...
- 15.6k
2
votes
0
answers
94
views
On fractional parts and Behrend’s construction
Given $\theta \in \Bbb{T}^D := \Bbb{R}^D/\Bbb{Z}^D$, write $f_\theta$ for the homomorphism from $\Bbb{Z}\to \Bbb{T}^D$ induced by $1\mapsto \theta$.
For $x\in \Bbb{T}^D$, let $||x||$ be the smallest $\...
- 3,499
2
votes
1
answer
402
views
Sets with certain property concerning density of sumsets
I am working with subsets of $[n]$ of the form $(A+B)\cap A$, where $A+B$ is a sumset. Namely, I am interested if there are nonempty sets $B$ such that whenever $A$ covers a positive proportion of $[n]...
- 807
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 ...
- 8,998
4
votes
0
answers
157
views
Multidimensional van der Waerden, bounds for squares
Given $r$, let $f(r)$ be the smallest $N$ such that for any $r$-coloring $C:\{1,\dots,N\}^2 \to \{1,\dots,r\}$, there exists $x,y,d\neq 0$ such that $C((x,y)) = C((x+d,y))= C((x,y+d))=C((x+d,y+d))$.
I ...
- 3,499
1
vote
0
answers
162
views
Difficulty understanding a step in the proof of multiset version of Cauchy-Davenport Theorem
In a paper "G. Kós, L. Rónyai, Alon’s Nullstellensatz for multisets, Combinatorica, 32(5) (2012) 589-605", the authors prove a multiset version of the Cauchy-Davenport Theorem (please see ...
- 167
1
vote
1
answer
115
views
Cardinality of $\{ n_i + i^k: i \in \mathbb{N} \} \cap [1,T]$ where $\{n_i \}$ is all natural numbers in some order
Let $n_1, n_2, ...$ be a sequence of natural numbers such that $\{n_i: i \in \mathbb{N}\}$ as a set is all of natural numbers. Let $k$ be a positive integer. Is is possible to obtain a lower bound of ...
- 3,625
14
votes
2
answers
902
views
Sylvester–Gallai theorem for small sets in a finite field
The well-known Sylvester–Gallai Theorem states that a set of $n>2$ points in $R^2$ not all on a line contains two points such that the line passing through these two points does not contain a third ...
- 13k
1
vote
2
answers
204
views
Only trivial solutions to system of linear diophantine equations possibly related to hamiltonian cycles in graphs
This might be related to counting hamiltonian cycles.
@Peter Taylor gave negative result about the one dimensional case, but we believe his attack is
not directly applicable to this question.
Given ...
- 25.4k
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{...
- 25.4k
9
votes
0
answers
265
views
If $A+A+A$ contains the extremes, does it contain the middle?
Let $b \ge 1$ and $A\subseteq [0,b]$ be a set of integers (all intervals will be of integers).
Write $hA := \underbrace{A + \ldots + A}_{h\text{ summands}} = \{ \sum_{i=1}^h a_i ~|~a_i \in A,\, \...
- 825
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 ...
- 13.4k
21
votes
1
answer
739
views
Does $A-A=\mathbb Q$ hold for $A=\{x^4+y^4:\ x,y\in\mathbb Q\}$?
Let $A=\{x^4+y^4:\ x,y\in\mathbb Q\}$. Then
$$A-A:=\{a-b:\ a,b\in A\}=\{u^4+v^4-x^4-y^4:\ u,v,x,y\in\mathbb Q\}.$$
Motivated by Question 415482, here I ask the following question.
Question. Is it true ...
- 15.6k
4
votes
0
answers
154
views
How many convex or concave subsets are contained in an arbitrary set of $n$ real numbers?
This question is closely related to this post. A set $A=\{a_1<a_2<\dots<a_n\} \subset \mathbb R$ is said to be convex if the consecutive differences are non-decreasing, i.e. if $a_{j+1} - a_j ...
- 1,498
6
votes
0
answers
491
views
At most two elements give 1 to n
Fix a positive integer $m$. Let $n$ ( $= n(m)$) be the largest positive integer for which there exists some subset $\{a_1,\ldots,a_m\} \subseteq \{1,2,\ldots,n\}$ of $m$ positive integers between $1$ ...
- 165
2
votes
1
answer
213
views
Coefficient of a term in a several variable polynomial multipled with Vandermonde determinant
Let $\Delta_n(x_1, \ldots, x_n)$ denote the Vandermonde determinant $\displaystyle \prod_{1 \leq i < j \leq n}(x_j - x_i)$. Let $c_1, \ldots, c_n$ and $K$ be nonnegative integers satisfying
$$c_1 + ...
- 167
5
votes
2
answers
706
views
Distribution of some sums modulo p
Fix a finite set of integers $S$ and a prime number $p$. Let $(a_1, a_2, \dotsc, a_n)$, $(b_1, b_2, b_3, \dotsc, b_n)$ be two sequences of integers where the numbers $a_i$ and $b_i$ are chosen ...
- 1,101
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 ...
- 1,101
1
vote
0
answers
175
views
For any $n-1$ elements of $\mathbb Z/n\mathbb Z$, we can make $0$ using $\{-,+,\times\}$ without parentheses
MSE: Just using $+$ ,$-$, $\times$ using any given n-1 integers, can we make a number divisible by n? (no brackets allowed)
Is there any hope in proving the following? (Cross-posted here after a ...
- 611
0
votes
0
answers
266
views
Is there a permutation $\tau\in S_n$ with $\tau(1)^{\tau(2)}+\cdots+\tau(n-1)^{\tau(n)}+\tau(n)^{\tau(1)}$ a square?
Let $n>1$ be an integer, and let $S_n$ be the symmetric group of all the permutatins of $\{1,\ldots,n\}$.
I'm curious whether there is a permutation $\tau\in S_n$ such that
$$\tau(1)^{\tau(2)}+\...
- 15.6k
10
votes
2
answers
926
views
Converse to Erdős' conjecture on arithmetic progressions
I apologise in advance if this has been asked here before. I did a search and did not find anything obvious. Erdős' conjecture states that if $A\subseteq {\bf N}$ is such that $\sum_{n\in A} n^{-1}$ ...
- 807
3
votes
0
answers
179
views
Ellenberg and Gijswijt's result on arithmetic progressions in subsets of $\mathbb{F}_q^n$ and a generalisation to sets of linear equations
Ellenberg and Gijswijt showed that the largest subset of $\mathbb{F}_q^n$ with no three terms in arithmetic progression has size $c^n$ where $c<q$.
Ellenberg and Gijswijt actually proved a ...
- 4,862
5
votes
1
answer
218
views
Computational version of inverse sumset question
Let $p$ be prime and $\mathbb{F}_p$ the finite field with $p$ elements. Suppose we have a set $B\subseteq \mathbb{F}_p$ satisfying $|B|<p^{\alpha}$ for some $0<\alpha<1$ and there exists $A\...
user156885
1
vote
1
answer
218
views
Average size of iterated sumset modulo $p-1$,
Given a prime $p$, what is the average size of the iterated sumset, $|kA|$, modulo $p-1$, with $p$ a prime, and $k$ given, with $A$ chosen at random?
You can pick any type of prime you like for $p$, ...
- 221
6
votes
4
answers
627
views
Request for an exact formula related to a partition in number theory
The Frobenius equation is the Diophantine equation $$
a_1 x_1+\dots+a_n x_n=b,$$
where the $a_j$ are positive integers, $b$ is an integer, and a solution $$(x_1, \dots, x_n)$$
must consist of non-...
- 10.5k
4
votes
0
answers
194
views
A conjecture on the cardinality of minimal mediated sequences
For a sequence of integer numbers $A=\{0,q_1,\ldots,q_m,p\}$ (arranged from small to large), if every $q_i$ is an average of two distinct numbers in $A$, then we say $A$ is a mediated sequence.
...
- 133
19
votes
4
answers
865
views
Size of sets with complete double
Let $[n]$ denote the set $\{0,1,...,n\}$. A subset $S\subseteq [n]$ is said to have complete double if $S+S=[2n]$. Let $m(n)$ be the smallest size of a subset of $[n]$ with complete double. My ...
- 30.5k
3
votes
0
answers
147
views
Under what conditions on $A$ and $v$ is the size of the sumset $v \cdot A + A$ over $\mathbb{F}_p$ equal or close to $|A|^2$?
Let $p$ be a prime, let $A$ be a subset of $\mathbb{F}_p$, and let $v \in \mathbb{F}_p \setminus \{0\}$.
Under what conditions is $|v \cdot A + A|$ (that is, $|\{ va + b : a \in A,\ b \in A \}|$) ...
10
votes
2
answers
676
views
Sets A such that A+A contains the largest set [0,1,..,t]
I look for a reference for the following problem.
Given an integer $k$, find a set $A\subset\mathbb{N}$ with $|A|=k$
that maximizes $t$ such that $\left[0,1,..,t\right]\subset A+A$.
- 572
2
votes
2
answers
293
views
Calculating the number of solutions of integer linear equations
Let $N$ be a natural number. Consider the following set of matrices whose entries are non-negative integers:
$$X_N:=\left\{(c_{ij})_{i,j=1}^4\in M_4(\mathbb{Z}_{\geq 0})\bigg| \sum_j c_{1j} = \sum_i ...
- 5,039
13
votes
2
answers
2k
views
Positive integers written as $\binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8$ with $w,x,y,z\in\{2,3,\ldots\}$
Let $\mathbb N=\{0,1,2,\ldots\}$. Recall that the triangular numbers are those natural numbers
$$T_x=\frac {x(x+1)}2\quad \text{with}\ x\in\mathbb N.$$
As $T_x=\binom{x+1}2$, Gauss' triangular number ...
- 15.6k
2
votes
0
answers
205
views
Concerning the identity in sums of Binomial coefficients [closed]
Consider the following identity
$$\sum_{k=1}^{n}\binom{k}{2}=\sum_{k=0}^{n-1}\binom{k+1}{2}=\sum_{k=1}^{n}k(n-k)=\sum_{k=0}^{n-1}k(n-k)=\frac16(n+1)(n-1)n$$
As we can see the partial sums of binomial ...
- 167
0
votes
1
answer
178
views
A question on "SUM-PRODUCT...VIA KLOOSTERMANN SUMS", by Hart, Iosevich and Solymosi
In this paper https://arxiv.org/pdf/math/0609426.pdf, the authors, state, as a consequence of Theorem 1.1, the following sum-product estimate.
Theorem 1.1 says that for all $A\subset\mathbb{F}_q$, we ...
- 311
1
vote
1
answer
261
views
Largest cardinality $n$ of a subset $A$ of $\{1,2,\ldots,M\}$ such that $(A+A) \cap A$ is empty
Assume that the set $A$ does not have simple structures (such as the case that when all elements are odd numbers in $[1,M/2]$ then all sums are even thus there are no solutions, as pointed out by @...
- 10.4k
3
votes
0
answers
282
views
A new combinatorial problem for finite groups
In a recent preprint arXiv:1811.10503, I proved that if $a_1,\ldots,a_n$ are distinct elements of a torsion-free additive abelian group $G$, then there is a permutation $\pi\in S_n$ such that all ...
- 15.6k
0
votes
1
answer
298
views
Discrepancy in non-homogeneous arithmetic progressions
I have a doubt, Roth's discrepancy theorem [1] says that there is a subset of arithmetic progressions $A \in [n]$ where any function $f:N\rightarrow \left \{ -1,1 \right \} $ implies
$ \left | \...
1
vote
0
answers
154
views
On the set $\{\sum_{k=1}^n \lambda_ka_k:\ a_1,\ldots,a_k\ \text{are distinct elements of}\ A\}$
For a field $F$ let $p(F)=p$ if the characteristic of $F$ is a prime $p$, and $p(F)=+\infty$ if $F$ is of characteristic zero.
In 2007 I considered the linear extension of the Erdos-Heilbronn ...
- 15.6k
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,...
- 15.6k