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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3
votes
0answers
138 views
+50

On covering by smooth numbers

Denote $P(n)=\mathsf{greatest}\mbox{ }\mathsf{prime}\mbox{ }\mathsf{factor}\mbox{ }\mathsf{of}\mbox{ }n$. Denote $S(x,y)=\{n<x: P(n)<y\}$. Denote $S_t(x,y)=\sum_{i=1}^tS(x,y)$ as $t$-fold ...
-1
votes
0answers
56 views

A finite field $0$-free restrictive sumset problem

Fix $m\in\Bbb N$. Let $\Bbb F_2^{\big\lceil\log^d m\big\rceil}$ be $\Big\lceil\log^dm\Big\rceil$-dimensional vector space over $\Bbb F_2$. What is minimum $d$ (if it exists - need sufficient ...
17
votes
0answers
381 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 ...
4
votes
0answers
190 views

Area defined with $\pm$ closedness

Denote $B_n\subset\Bbb R^n$ to be unit ball at origin. Denote $S\subset B_n$ to region of type $\mathsf I$ if it satisfies $$s\in S\iff\forall t\in S, s+t\in S\mbox{ or }s-t\in S$$ I am convinced ...
4
votes
0answers
110 views

Additivity of upper densities with respect to arithmetic progressions of integers

Let $\mathsf{d}^\star$ be the asymptotic upper density, defined on the power set of positive integers $\mathbf{N}^+$, so that $$ \mathsf{d}^\star\colon \mathcal{P}(\mathbf{N}^+) \to\mathbf{R}\colon ...
35
votes
3answers
3k 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 $o(n^2)$?
3
votes
1answer
656 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 ...
7
votes
1answer
188 views

Iterated sumset inequalities in cancellative semigroups

This question is motivated by the following well-known theorems: Thm (Plünnecke): If $A$ is a finite nonempty subset of an abelian group, then for every $n$ we have $|A^n| \le ...
6
votes
1answer
208 views

What pairs of sets have additive energy?

In an abelian group, the additive energy between two sets is $$E(A,B)=|\{(a_1,a_2,b_1,b_2) \in A\times A\times B\times B:a_1+b_1=a_2+b_2\}|$$ which is ranges from $|A||B|$ to $(|A||B|)^{3/2}$. What I ...
2
votes
1answer
66 views

Elaboration of a certain section of a paper by Thanigasalam

In section 11 of this paper by Thanigasalam, it says "... we get $G(10)\le 105$, and this implies that $H(10) \le 107$". However, it is very unclear how this follows. Why is it the case that $G(10)\le ...
0
votes
0answers
89 views

Determing signs of Taylor coefficients in entire functions

This is a continuation of Determining when combinatorial sums are zero Suppose $f(x)$ is an entire function approximated by polynomials with only negative real zeros. Suppose further that ...
6
votes
2answers
261 views

Determining when combinatorial sums are zero

To keep things simple with a specific example, we ask: Prove that $\displaystyle\ a_n:=\frac{1}{n!}\sum_{k=0}^n \binom{n}{k} \frac{1}{k!} (-1)^{n-k}$ is zero if and only if $n=1$. (Or find a ...
1
vote
1answer
98 views

On the upper Banach density of the set of positive integers whose base-$b$ representation misses at least one prescribed digit

Let $b$ be a fixed integer $\ge 2$ and $A$ a proper subset of $\{0, \ldots, b-1\}$. Then define $X$ to be the set of all positive integers whose base-$b$ representation consists only of digits from ...
7
votes
1answer
281 views

Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natural density of $X$ is zero?

It is a result from additive-theory folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. ...
6
votes
0answers
143 views

Distribution of trivial subset sums

Suppose I have a set $S$ of $n$ integers picked independently, uniformly at random from $[-L, L].$ Let $z(S)$ be the number of subsets of $S$ which sum to zero. The question is: what is the ...
3
votes
0answers
128 views

Goldbach's problem in algebraic number fields [duplicate]

Are there any results on the representation of numbers in algebraic number fields as the sum of primes in the ring of integers in that field? There are some results for Waring's problem in other ...
5
votes
3answers
505 views

Any rigorous way to claim that sums with repeat summands are few?

Let $B \subset \mathbb{Z}^+$. Define $r_{B,h}(n)$ to be the number of ways of writing $n$ as the sum of $h$ elements of $B$ and $R_{B,h}(n)$ the number of ways to write $n$ as the sum of $h$ DISTINCT ...
3
votes
1answer
174 views

When are the powers of 2 sum-free mod n?

I've encountered the following question in my research: Let $A$ be a subset of $\mathbb{Z}/n\mathbb{Z}$. Let me call $A$ "sum-free" if there is no solution to $x+y=z$ for $x,y,z \in A$ with distinct ...
3
votes
1answer
155 views

higher dimensional analogue of EGZ theorem

The EGZ theorem states that any multiset of $2n-1$ integers has a subset of size $n$ the sum of whose elements is a multiple of $n$. Kemnitz-Reiher theorem is a 2-dimensional analogue of EGZ. Here is ...
4
votes
1answer
219 views

Ref. request: Additive probability measure on $\mathcal P({\bf N})$ supplies subset of $\mathbf R$ without Baire property

ZFC proves, among the other things, the existence of a (finitely) additive probability measure $\theta: \mathcal P(\mathbf N) \to \mathbf R$ on the power set of $\mathbf N$ such that $\theta(X) = 0$ ...
4
votes
1answer
155 views

Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathbf N^+$

Let $\mathcal D$ be the set of all (finitely) additive probability measures $\mu^\ast: \mathcal P(\mathbf N^+) \to [0,\infty[$ such that $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$ for all $X ...
1
vote
0answers
54 views

Does there exist $k\ge2$ s.t. $X \subseteq \mathbf N^+$ has positive upper Banach density if the counting function of $X$ is $\gg n/\log^{[k]}(n)$?

Does there exist an integer $k \ge 1$ such that ${\sf bd}^\ast(X) > 0$ whenever $X \subseteq \mathbf N^+$ and $\pi_X(n) \gg \frac{n}{\log^{[k]}(n)}$ as $n \to \infty$? Here, ${\sf bd}^\ast$ is the ...
3
votes
2answers
346 views

Who needs a symmetric upper asymptotic density on the integers?

The upper asymptotic density on $\mathbf Z$, viz. the function $$ {\sf d}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [1,n]|}{n}, $$ has a ''symmetric ...
2
votes
1answer
106 views

Representation numbers of numerical semigroups

I've been playing around with numerical semigroups lately. I'm pretty new to this stuff, so I apologize in advance if my notation is non-standard. Fix positive integers $x_1,\dots,x_r$ with ...
4
votes
1answer
114 views

Approximate homomorphisms

Let $f:Z_2^n \to Z_p$ be a one-to-one map, where say $2^n<p<2^{n+1}$. What is the maximal probability that $\Pr[f(x+y)=f(x)+f(y)]$ where $x,y \in Z_2^n$ are uniform and independent? The identity ...
4
votes
0answers
87 views

Sparsifiers for 3-term arithmetic progressions

Let $G$ be a finite abelian group of odd order, let $D\subseteq G$, and $\epsilon \in (0,1)$. For $S\subseteq G$ define $$ \Lambda(S) = \frac{1}{|S||G|} \sum_{s\in S}\sum_{g\in G} ...
9
votes
2answers
363 views

Sumsets and dilates: does $|A+\lambda A|<|A+A|$ ever hold?

The following problem is somehow hidden in this recently asked question, but I believe that it deserves to be asked explicitly. Is it true that for any finite set $A$ of real numbers, and any real ...
1
vote
1answer
194 views

Ask the name of a combinatorial theorem

It is a classical theorem. For given integer $n \ge 1$, among ${n\choose{n/2}} = 2^{(1-o(1)n)}$ strings in the cube $\{0, 1\}^n$ with weights $n/2$, i.e., $n/2$ indices are 1, there are at least ...
18
votes
0answers
405 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 ...
3
votes
0answers
89 views

Reference for a lemma on the asymptotic upper density of special sets with large gaps and intervals

Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii). In a joint paper that I am ...
5
votes
2answers
202 views

Ordered lattice point enumeration

I initially asked this question over at StackOverflow as it has algorithmic flavor to it, but I haven't been getting much traction so I thought I would probe the mathematics community. Setup: Let ...
13
votes
1answer
383 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 ...
6
votes
2answers
207 views

What methods do we have to understand the spectrum of matrices with restricted entries?

Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"), What is the largest possible spectral radius of a $n \times n$ matrix ...
0
votes
1answer
69 views

What is the maximal number of solutions of $\sum_{i = 1}^n 1/a_i^x - \sum_{i = 1}^m 1/b_i^x = 0$?

What is the maximal number of solutions of the following equation? $\sum_{i = 1}^n 1/a_i^x - \sum_{i = 1}^m 1/b_i^x = 0$ where $x$ is the unknown and $n, m$, $a_i$'s, $b_i$'s are constant. It ...
6
votes
3answers
423 views

Conditions for an analogue of Cauchy-Davenport for simple groups

What conditions would be sufficient for a generalization of Cauchy-Davenport for simple groups? I can see two possible difficulties with a generalization for general groups: The sets could both be ...
0
votes
0answers
109 views

Generalized arithmetic progressions contained in Bohr sets

Recall that generalized arithmetic progression of dimension $d$ is by definition a set of the form $P = P_1+\dots+P_d$, where $P_j = \{lp_j\ \mid \ |l|\leq l_j\}\subset \mathbb Z$ is an ordinary ...
85
votes
7answers
4k 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 ...
12
votes
2answers
769 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 ...
24
votes
3answers
543 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 ...
5
votes
1answer
195 views

Element with unique representation in A+B

Let $A, B \subseteq \mathbb{Z}$ be finite subsets of the integers. Then there exists an element in $A+B$ with a unique representation as a sum of an element in $A$ and an element in $B$, namely ...
2
votes
2answers
518 views

Asymptotic formula for the number of ways to write a number as the sum of $k$ triangular numbers

How would one derive an asymptotic formula for the number of representations of a number $n$ as the sum of $k$ numbers of the form $\frac{m(m + 1)}{2}$ I think that one could use the circle method, ...
2
votes
2answers
202 views

Applications of Szemeredi's Theorem

Szemeredi's Theorem is a famous theorem in Additive Combinatorics, Ergodic Theory and maybe some other parts of Mathemtaitcs: (Szemeredi's Theorem) Let $\Lambda \in \mathbb{Z}$ be a subset of ...
2
votes
0answers
54 views

How is the structure of spectrum in cap-sets with no strong increments unrealistic if density is too large?

I am reading this excellent paper by Bateman and Katz on improved bounds on the cap set problem. Let A be a set in $\mathbb{F}_3^n$ containing no 3-term arithmetic progression and let $A(x)$ denote ...
1
vote
1answer
115 views

Pollard's inequality modulo a composite number

Here is question similar to another one asked today (Proof of Pollard's inequality). When working in $\mathbb{Z}/n\mathbb{Z}$ for composite $n$ Cauchy-Davenport and Pollard's inequalities may not ...
2
votes
2answers
170 views

Proof of Pollard's inequality

Let $A, B \subseteq \mathbb{Z}_p$, $p$ prime, $|B| \leq |A|$. If $N_t$ denotes the number of elements of $\mathbb{Z}_p$ having at least $t$ representations as $a+b$, $a \in A, b \in B$, Pollard's ...
2
votes
1answer
110 views

On a problem about $GF(2)^n$

For $A\subseteq {\mathbb F}_2^n$ let $$ Q(A)=\{\alpha+\beta\mid \alpha,\beta \in A,\ \alpha\neq\beta \}. $$ I want to prove or disprove that if $|A|=2^k+1$ for some integer $k$, then $$ ...
5
votes
3answers
292 views

Sets of natural numbers with finite intersections and divergent sums of reciprocals

Does there exist an uncountable collection $\Lambda$ of infinite subsets of the set of natural numbers such that (i) any two distinct subsets in the collection have a finite intersection and (ii) the ...
2
votes
0answers
103 views

Applications of Freiman's theorem?

What are some interesting applications of Freiman's theorem or, better-yet, its recent generalizations (eg Green-Ruzsa) that could be included in a graduate course in additive combinatorics? I'm ...
8
votes
2answers
594 views

Most dense subset of numbers that avoids arbitrarily long arithmetic progressions

The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression. I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid ...
3
votes
0answers
233 views

Solving a doubly exponential generating function

I am analyzing the average time complexity of some algorithm on some probabilistic model, and I've come to a doubly exponential sequence for which I cannot find corresponding generating function. I ...