Questions on the subject additive combinatorics, also known as arithmetic combinatorics, such as questions on: additive bases, sum sets, inverse sum set theorems, sets with small doubling, Sidon sets, Szemerédi's theorem and its ramifications, Gowers uniformity norms, etc. Often combined with the ...
1
vote
1answer
210 views
+50
Valid 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 = 0, 1, 2, ...
-1
votes
0answers
11 views
Learning roadmap for additive combinatorics [migrated]
I have read Calculus by Michael Spivak. Now I want to learn additive combinatorics though I have no experience with combinatorics or probability theory. To my understanding, there is a book on the ...
2
votes
1answer
237 views
A reference for this possibly well-known fact concerning the Kakeya conjecture?
I believe I have read or heard somewhere that the Kakeya conjecture would follow from appropriate lower bounds for the minimal size of a subset of $\{ 1 , \cdots , N\}$ which contains a translate of ...
1
vote
1answer
114 views
Does the asymptotic formula for Partitions into parts <c exist?
A partition of $n$ is a weakly decreasing tuple of numbers $(\lambda_1,\lambda_2,\lambda_3,....\lambda_k)$ whose sum is $n.$ A natural problem studied is counting partitions whose summands ...
1
vote
2answers
60 views
Generalization of Rado's Single Equation theorem
Yesterday I came up with a problem: Can we color each point of the plane with finitely many colors such that there doesn't exist any monochromatic regular polygonal?
But I found the problem is too ...
0
votes
0answers
51 views
Greedy sequences without k-term arithmetic progressions
If $S_k$ is the greedy sequence with no length-k arithmetic subsequence, (ie $S_3$ = A003278 , $S_4$ = A005837 , $S_5$ = A020655 ), is it guaranteed that any other sequence $a$ with no length-k ...
4
votes
1answer
320 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 ...
3
votes
1answer
125 views
Non-asymptotically densest progression-free sets
For the context of this question, a progression-free set is a subset of integers that does not contain length-three arithmetic progressions.
For large $N$, it is known that $[N] = \{1, \ldots, N\}$ ...
2
votes
1answer
121 views
Set of small numbers with distinct $k$-sums
Let $A$ be a set of $n$ positive integers with distinct $k$-sums. In other words, if $a_1\le\cdots\le a_k$ and $b_1\le\cdots\le b_k$ are elements of $A$ such that $a_1+\cdots+a_k=b_1+\cdots+b_k$, then ...
1
vote
1answer
219 views
Karolyi's theorem for finite groups and its extensions
Suppose that $\mathbb A = (A, +)$ is a (possibly non-commutative) group, and denote by $p(\mathbb A)$ the minimum of $|S|$ as $S$ ranges in the set of non-trivial subgroups of $\mathbb A$, with the ...
3
votes
0answers
69 views
Closest sumset to a set
Suppose $A$ is a subset of the finite field with $p$ elements. What is the best approximation of $A$ by a sumset $B+C$ in the sense that $|A\Delta (B+C)|$ is minimal? Of course if $B=A-x$ and ...
20
votes
2answers
709 views
The Erdős-Turán conjecture or the Erdős' conjecture?
This has been bothering me for a while, and I can't seem to find any definitive answer. The following conjecture is well known in additive combinatorics:
Conjecture: If $A\subset \mathbb{N}$ ...
2
votes
2answers
92 views
A measure of closure under sumset?
Let $G$ be an Abelian group. Let $A \subseteq G$. In additive combinatorics, one of the primary measures of the additive structure of $A$ is its additive energy, defined as $E(A) = ...
3
votes
2answers
245 views
Average involving the Euler phi function
Does
$$\frac{1}{N^2}\sum _{d=1}^N \log d \sum _{n=1}^{N/d} \frac{\phi(n)}{\log (dn)},$$
converges or not when $N$ goes to infinity?
1
vote
0answers
57 views
Distribution of colors in the number of integer partitions of n
Given an integer $n$ the number of partitions of $n$ into two colors can be represented as
$$p_2(n)=\sum_{k=0}^n p(k)p(n-k)$$ where $p(k)$ counts the number of ordinary partitions of $k.$ What is the ...
5
votes
1answer
275 views
Additive Combinatorics - reference request
Let $A$ be a finite set of integers with $|A \hat{+} A| \leq K|A|$, where the $\hat{+}$ denotes restricted sumset: the set of all $a_1 + a_2$ with $a_1, a_2 \in A$ and $a_1 \neq a_2$.
Claim: $|A + ...
1
vote
0answers
130 views
Estimates for the size of the product set [n].[n] [duplicate]
Possible Duplicate:
Number of elements in the set {1,…,n}*{1,..,n}
Writing $[n]$ for the set $\lbrace1,2,...,n\rbrace$, let $P_n$ denote the product set $[n].[n]$, i.e.
$$ P_n = ...
0
votes
2answers
233 views
Solution in distinct elements for a system of $n$ equations over finite fields
The following problem is motivated by pure curiosity; it is not a part of any research project and I do not have any applications.
Problem:
Let $\{y_1 , y_2 , \dots, y_n \}$ be arbitrary (distinct) ...
12
votes
1answer
492 views
On the $L^1$-norm of certain exponential sums.
I am stuck with an elementary-looking problem, which does not belong to my usual field of research so I eventually decided to ask it on MO.
Let $S$ be a finite set of integers. For $P$ a subset of ...
2
votes
1answer
112 views
existence of arithmetic progression of nonzero density
This is a stronger version to Szemerédi's theorem.
Let $C : \mathbb{N}\rightarrow 2^{\mathbb{N}}$ be a choice function such that $C(n)$ is a subset of $\{1,...,n\}$ with size at least $\frac{n}{M}$ ...
0
votes
0answers
28 views
Minimum number of solutions in a system of equalities and non-equalities
Let $k<N$ and $P_1, ..., P_{2k+1}, \lambda_1, ..., \lambda_k$ be elements of a finite group $G$ of size $N$.
Find the minimum number of solution of the system
$$P_{2i} + P_{2i+1} = \lambda_i, ...
1
vote
1answer
301 views
About a sumset in $\mathbb{Z}_{2k}$
Suppose $U\subset\mathbb{Z}_{2k}$ with $|U|=k$. Let $U^c$ denote the complement of $U$.
Let $v\in \mathbb{Z}_{2k}^{\times}$. How much is it known about $U+vU^c$?
For example: When $U+v\complement U ...
3
votes
0answers
86 views
Bounds on difference sets of relatively dense A \subseteq {1, …, n}
Let $A \subseteq \{1, \dots, n\}$ and let $A-A = \{a-b | a,b \in A\}$. Is it possible to obtain a general bound of the form $|A - A| = O(n^\beta)$ for some $\beta < 1$? If not, can something like ...
4
votes
3answers
315 views
Selecting $k$ integers from an interval $[0, N]$ to maximize the minimum difference between pairwise sums
I have an optimization problem where I need to select $k$ integers over the interval $[0, N]$ s.t. I maximize the minimum difference between any pairwise sum of the $k$ integers (where we also include ...
13
votes
0answers
270 views
probability of zero subset sum
Almost 17 years ago, I asked the following question on USENET, motivated by a method in numerology (I kid you not).
Pick integers $n \ge 2$, $k \ge 1$. Toss $n$ $k$-sided dice. The sides of each die ...
1
vote
1answer
231 views
Generalizations of Cauchy-Davenport Theorem
The Cauchy-Davenport Theorem says that if $A_1, \ldots, A_k$ are subsets of ${\mathbb Z}_p$, $p$ prime, then $| \sum_i A_i | \geq \min (p, \sum_i |A_i| -k +1)$.
I am looking for a generalization that ...
0
votes
1answer
73 views
sum of a binomial-weighted entity
Does anybody know a closed-form solution for the following expression (N>=1)? I don't even know where to begin with the i+n denominator.
Sum of i=0 to n Combin(n,i) * (2i/(i+n)) / (2^n)
4
votes
0answers
264 views
Largest set of integers without 3-term arithmetic progressions mod $n$
I am interested in a sharp bound on the largest possible size $e_3({\boldsymbol{Z}_n})$ of a subset $S \subset \boldsymbol{Z}_n$ such that for any three distinct elements $a, b, c \in S$ we have $a+b ...
6
votes
1answer
223 views
Recovering Sidon sets from difference sets, part 2.
This is inspired by a recent question. A set $A \subset \mathbb{Z}/n\mathbb{Z}$ with $|A|=m$ is a Sidon set if all the pairwise sums of distinct elements are unequal: $A+A=\{a+a' \mid a,a' \in A, a ...
2
votes
2answers
276 views
Recovering Sidon sets from difference sets
How can I recover a Sidon set $A\subseteq \mathbb{Z}/n\mathbb{Z}$ from the set $A-A\subseteq \mathbb{Z}/n\mathbb{Z}$?
Is it even unique? (up to translation and reflection)
($A-A$ stands for the set ...
