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 ...

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163 views

Coin problem with permutations

Let $a,b,c$ be positive integers with gcd$(a,b,c)=1$, and let $\mathbb{N}$ denote the set of nonnegative integers. It is well known that $\mathbb{N} \setminus (a \mathbb{N}+b \mathbb{N} + c ...
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330 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 ...
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334 views

Correlation of Fourier transforms of characteristic functions

Let $A$ and $B$ ($A\subset B$) be subsets of a finite abelian group $G$. (For the sake of argument, you can take $G$ to be $\mathbb{Z}/p\mathbb{Z}$ for large $p$, say.) Write $1_S$ for the ...
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342 views

What are the limits of the Erdős-Rankin method for covering intervals by arithmetic progressions?

To construct gaps between primes which are marginally larger than average, Erdős and Rankin covered an interval $[1,y]$ with arithmetic progressions with prime differences. A nice short exposition is ...
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115 views

Erdös-Fuchs Theorem for multivariate linear forms

Let $A$ be an infinite set of positive integers, and denote by $r(n)$ the number of solutions to the equation $a+a'=n$, with $a,\, a' \in A$. It is not very difficult to show that if $r(n) > 0$ ...
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484 views

Cliques in the Paley graph and a problem of Sarkozy

The following question is motivated by pure curiosity; it is not a part of any research project and I do not have any applications. The question comes as an interpolation between two notoriously ...
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300 views

Small maximal sets with no 3-AP?

Everyone knows about the problem of finding the largest set in 1,2,...,N which contains no arithmetic progression of length 3.....what happens if we look at the set which is the smallest maximal ...
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124 views

Distribution of subset-sums

Let $A$ be a set of $n$ integers uniformly distributed in $\{0,\dots,N-1\}$. Let $S$ be the set of subset-sums modulo $N$ of $A$. Let $f_{n,N}(k)$ be the probability that $|S|=k$. Is there an ...
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521 views

Expectation of Gowers norm

This was a problem that came up during a course on Analytic Combinatorics that I had taken this semester. Here's the problem: Let $\mathbf{F}$ be the set of boolean functions, $f: \mathbb{F}_2^n ...
4
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115 views

Other applications of the 'increment' approach

I would like to hear about other instances of the so-called 'increments' approach, first used by Roth to prove that a subset of $\mathbb{N}$ of positive upper density contained infinitely many ...
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321 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 ...
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85 views

Number cubes with consecutive line sums

This is barely of research interest, but I'd classify it as a curiosity with connections to combinatorics. The problem is to place integers in an $n \times n \times n$ array so that all $3n^2$ line ...
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55 views

Distribution of the evaluation (at a non-trivial root of -1) of polynomials with small coefficients

When we studied some cryptographic protocol, we came accross the following problem, which seems linked to the uniformity of the residues of small multiplicative subgroups of $\mathbb{F}_q$. Problem ...
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101 views

Doubling for Sumset of the same set

Let $G$ ($G=\mathbb{Z}^n_2$ for my case) be a additive group and $A$ be a subset of $G$. For any set $S\subseteq G$ define its doubling as $$\sigma (S)=\dfrac{|S+S|}{|S|}$$ Suppose $A$ has small ...
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89 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 ...
3
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0answers
102 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 ...
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188 views

Convex subsets of sumsets

There's a famous old problem in additive number theory which asks whether, for any $\epsilon > 0$, if $B$ is a 2-basis for the set $A =$ {$1^2,2^2,...,n^2$}, i.e.: a set for which $B+B \supseteq ...
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109 views

Other interesting examples of pseudo-random measures

Here we use the notion of pseudo-randomness (or variant of) used in the proof of the Green-Tao theorem. In particular, we say a measure $\nu : \mathbb{Z}_N \rightarrow \mathbb{R}^+$ is pseudo-random ...
3
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304 views

A question about Erdos thin bases

Let $B \subset \mathbb{N}$ be an additive basis of order $h$. Define $r_{B,h}(n)$ to be the number of ways $n$ can be written as a sum of $h$ elements of $B$. In particular, $B$ is a basis of order ...
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61 views

“Pseudo-random” subsets of additive bases

We say that a subset $B \subset \mathbb{N}$ is an (asymptotic) additive basis of order $k$ if the set $kB = B + \cdots + B = \mathbb{N} \setminus C$, where $C$ is a finite set of positive integers. ...
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106 views

Lower bound on P(Y > 3/2 E(Y)) where Y = # of triangles in graph

Context This is exercise 1.7.1 of Tao/Vu's Additive Combinatorics, generally considered a graduate level math textbook. My question is not "How do I solve this exercise?" My question is "What is ...
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0answers
241 views

Prime divisors of the difference set

Fix $c\in(0,1)$, and let $N$ be a (large) positive integer. Given a set $A=\{0=a_1<\dots<a_n=N\}$ of density $\alpha:=n/N>c$ with $\gcd(A)=1$, I want to find a prime dividing as few ...
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909 views

Fun question in additive combinatorics

It is easy to see that for a finite set of integers $A$ of cardinality $n$, the cardinality of the sumset $A+A$ satisfies $$ 2n-1\leq |A+A|\leq \frac{n(n+1)}{2}. $$ The lower bound is essentially ...
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56 views

Stronger condition than being a normal polytope?

A polytope $P$ with integer vertices is called normal if for every $p = \sum_j a_j p_j $ such that $a_j \geq 0$, $\sum_j a_j = k \in \mathbb{N}$, $p_j$ are vertices of $P$ and $p$ is an integer ...
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123 views

Finite fields: alternating sums of values of polynomials

Notation In what follows let $p$ be a (odd, if needed) prime, $e$ a positive integer, $q = p^e$; $\mathbb{F}_q$ will denote a finite field with $q$ elements whose prime subfield will be denoted as ...
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293 views

Green-Tao style theorem for quadratic regressions (Ulam Spiral)

This is a naive question about number theory. Looking at an Ulam spiral which illustrates primes of the form e.g. $4x^2-2x+c$ and other quadratic equations $ax^2+bx+c$, with $c>0$, there appears a ...
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0answers
86 views

Examples of using additive structure to analyze size of a set mod some ideal?

There are examples of proofs that analyze the structure of some set $S$ of integers by looking at the size of the set mod some positive integer $n$ (for example, the $3k-3$ theorem.) Are there ...
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40 views

Collecting terms of a linear expression with nested sums and combinatorics in coefficients

I need to collect the $\Pr(\cdot)$ terms of the following expression: $\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left( 1-\theta \right) }\right) ^{m}}\left[ ...
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55 views

Cover one finite subset of integers by another one

Let $A$, $B$ be two finite subsets of integers. We denote by $C(A, B)$ the minimum number of shifts of $A$ to cover $B$. More formally, it can be written as $$ C(A, B)=\min\{|S|: S\subseteq ...
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0answers
56 views

$\sum_{a\in A} r(a)$, where $A\subseteq\mathbb{Z}_n$ and $r(a)$ is the number of representations of $a$ as sum of two elements from $A$

Reading Seva's answer to this question, I got lost at the line relating $\sum_{a\in A} r(a)$ to $|A\cap 2A|$. More precisely, restating the problem: Let $A\subseteq\mathbb{Z}_n$ be an set of ...
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0answers
120 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 ...
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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, ...
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129 views

upper bound on the size of sumset of lattice points

Let $\Lambda$ be a lattice (discrete additive subgroup) in $\mathbb R^n$ ($n\geq 2$). In my problem, $\Lambda$ lies in a $k$ dimensional ($1< k\leq n$) subspace of $\mathbb R^n$. Let $A\subset ...
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0answers
374 views

Calculating the Shapley value in a weighted voting game.

Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $ [q:1,1,1,1..1,2,2,..2] $. I need help with calculating the Shapley value of a player with a weight of $2$ and a ...
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0answers
242 views

Kakeya problem and arithmetic progressions

Here's something I also posted on Stackexchange recently. It's very related to the Kakeya problem, yet I fail to see why this is true. It goes like this: Let $r > 2$ be an integer parameter. Let ...