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Question 1

8votes

1answer

177views

Coefficients of $\prod\limits_{i=1}^k (1-z^{s_i})$: Additive combinatorics, inequality, or just simple observations?

Let $s_1$, $s_2$, ..., $s_k$ be $k$ nonzero integers (not necessarily positive or distinct). Let $P(z) := \prod\limits_{i=1}^k (1-z^{s_i})$. Prove that there exists $z$ on the unit circle (i.e., $|z| =...

Muses_China

1,108

askedNov 8 at 20:02

Question 2

7votes

2answers

354views

Finding a sum that is always divisible by $1+2+\dots+n$.

There are $𝑛$ consecutive integers $𝑚,𝑚+1,\dots,𝑚+𝑛−1$. Prove that you can choose some nonempty subset of these numbers whose sum is divisible by $1+2+\dots+𝑛 $.Edit: Okay, so here is what I ...

Obtuse

109

askedNov 8 at 15:55

Question 3

1vote

1answer

59views

Use the sampling argument to prove density Szemeredi (supersaturation) problem

This Exercise comes from Graph Theory and Additive Combinatorics (Yufei Zhao). I have tried for several days but end up with no idea.Exercise 1.3.7. (Density Szemeredi). Let $k\geq 3$. Assuming ...

zhangxm2312

105

askedNov 5 at 9:05

Question 4

12votes

2answers

285views

+100

Determining whether $A\subset X = \left\{0,1,\ldots,n\right\}$ can be written as $A=B+C$

Let $A$ be a nonempty subset of $X=\left\{0,1,\ldots,n\right\}$. We may ask whether $A$ can be written as $A=B+C$, where $B,C$ are also nonempty subsets of $X=\left\{0,1,\ldots,n\right\}$. Here, $B+C$ ...

Dilemian

1,341

askedOct 16 at 9:24

Question 5

3votes

1answer

76views

Understanding a step in Erdős's bound $2^{g(x)}\leq x\cdot g(x) \implies g(x)<\frac{\log{x}}{\log{2}} + [1+o(1)]\frac{\log{\log{x}}}{\log{2}}$

I'm looking at the problem 6. of an old Erdős paper Problems and results in additive number theory and I'm having trouble to understand a derivation he does.The problem starts withSome time ago ...

Patrick Scuracchio

33

askedSep 18 at 18:10

Question 6

0votes

0answers

72views

Sums of a quadratic-generated odd sequence and coverage of the even integers

For $\alpha > 0$, let$$A(\alpha) = \{ R_{\text{up-odd}}((\alpha x)^2 + \alpha x + 1) : x \in \mathbb{Z}_{\ge 0} \},$$where $R_{\text{up-odd}}(t)$ means: take the ceiling $\lceil t\rceil$; if it ...

Isaac Brenig

1,403

askedAug 15 at 15:38

Question 7

3votes

3answers

258views

Do the set of coprimes of a number form an additive basis?

For any positive integer $n,$ define $X:= \{ a: (a,n) = 1,\ 1\leq a < n \}.$ Let $(X+X)^*:= \{ (x+y)\pmod n: x,y\in X \}.$Conjecture: If $n$ is even, then $(X+X)^*= \{ 0,2,4,\ldots,n-2\}.$ If $n$ ...

Adam Rubinson

25.2k

askedJun 27 at 14:09

Question 8

2votes

0answers

149views

$n$ divisors of $n$ guarantees a subset sum of divisors adding up to $n$

Problem:Let $n$ be a positive integer, and let $D$ be a multiset of $n$ positive divisors of $n$. Prove that there must exist a subset of $D$, with sum equal to $n$.By multiset, I mean that, some ...

Sean

41

askedJun 27 at 9:19

Question 9

1vote

0answers

42views

Unsure about a step in the proof of Pollard's generalisation of the Cauchy-Davenport theorem

In J.M. Pollard's A generalisation of the theorem of Cauchy and Davenport (1974), Pollard shows that for a prime $p$ and $A,B \subseteq \mathbb{Z}/p\mathbb{Z}$, if $N_r$ denotes the number of elements ...

Mother_Worm

11

askedJun 17 at 12:32

Question 10

0votes

0answers

43views

Lattice of the sum of 2 elements belonging to a chain in a partially ordered finite abelian group

In a finite partially ordered abelian group, if we are given a chain $a<b<c<d...,$ we can construct a "generic" poset with only the necessary orders for the sum of $2$ distinct ...

pause lab

99

askedMay 26 at 2:53

Question 11

5votes

1answer

101views

3APs implies large additive energy

I am trying to solve Exercise 4.2 in Julia Wolf's Additive Combinatorics Notes. Let $G$ be any finite abelian group, but say $G = (\mathbb{Z} / 3\mathbb{Z})^n$ so that 3APs $(x, y, z)$ satisfy $x + y +...

ketsi

3,914

askedApr 22 at 14:11

Question 12

2votes

2answers

96views

How to show that a set in $\mathbb{F}_5^{2n}$ is $3$-progression-free

Let $$D = \left\{ (0, 0), (1, 0), (2, 0), (0, 1), (1, 1), (2, 1), (0, 2), (1, 2), (3, 2), (2, 3) \right\} \subset \mathbb{F}_5^2,$$let $ n $ be a multiple of $ \lvert D \rvert = 10 $, and define the ...

3nondatur

4,414

askedApr 8 at 21:37

Question 13

1vote

0answers

51views

Where is the assumption $0\in B$ used in Nathanson's generalization of Cauchy-Davenport (Theorem 2.1)?

I'm studying Theorem 2.1 from Additive Number Theory by Melvyn Nathanson (GTM 165), which extends the Cauchy-Davenport Theorem to composite modulus $n$, under the additional assumptions that:$0\in B$...

ltamb814

11

askedMar 31 at 15:24

Question 14

2votes

1answer

88views

How to generalise Behrend's construction to 4-APs?

In Behrend's classical paper a bound lower bound for the maximum size of $3$-free sets is constructed. I wonder, whether it is possible to generalise this approach to $4$-free sets.My thoughts on ...

3nondatur

4,414

askedMar 26 at 0:42

Question 15

0votes

0answers

43views

On counterexamples of the converse of Szmeredi's theorem

I am looking for a counterexample to prove that the converse of Szemeredi's theorem on arithmetic progressions does not hold. Whilst the Green-Tao theorem provides an interesting counterexample, I was ...

thomas martinelli

119

askedMar 24 at 17:17

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Questions tagged [additive-combinatorics]

Coefficients of $\prod\limits_{i=1}^k (1-z^{s_i})$: Additive combinatorics, inequality, or just simple observations?

Finding a sum that is always divisible by $1+2+\dots+n$.

Use the sampling argument to prove density Szemeredi (supersaturation) problem

Determining whether $A\subset X = \left\{0,1,\ldots,n\right\}$ can be written as $A=B+C$

Understanding a step in Erdős's bound $2^{g(x)}\leq x\cdot g(x) \implies g(x)<\frac{\log{x}}{\log{2}} + [1+o(1)]\frac{\log{\log{x}}}{\log{2}}$

Sums of a quadratic-generated odd sequence and coverage of the even integers

Do the set of coprimes of a number form an additive basis?

$n$ divisors of $n$ guarantees a subset sum of divisors adding up to $n$

Unsure about a step in the proof of Pollard's generalisation of the Cauchy-Davenport theorem

Lattice of the sum of 2 elements belonging to a chain in a partially ordered finite abelian group

3APs implies large additive energy

How to show that a set in $\mathbb{F}_5^{2n}$ is $3$-progression-free

Where is the assumption $0\in B$ used in Nathanson's generalization of Cauchy-Davenport (Theorem 2.1)?

How to generalise Behrend's construction to 4-APs?

On counterexamples of the converse of Szmeredi's theorem

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