**3**

votes

**1**answer

90 views

### 3-dimensional vectors satisfying certain equalities

Question: Are there 5 distinct vectors $u,v,w,x,y \in \mathbb{R}^3$, all on the unit sphere (i.e. $||u||=||v||=||w||=||x||=||y||=1$), such that:
$||u+v+x||=||u+v+y||=||u+w+x||=||u+w+y||=1$
?
Also, ...

**11**

votes

**1**answer

338 views

### A seemingly simple combinatorial object that must have an easy generating function

One more question related to my earlier "Special" meanders.
I am trying to isolate simplest problems related to it. Here is one.
For a composition (i. e. a tuple of natural numbers) ...

**7**

votes

**1**answer

316 views

### Does $|A+A|$ concentrate near its mean?

Fix $N$ to be a large prime. Let $A \subset \mathbb{Z}/N\mathbb{Z}$ be a random subset defined by $\mathbb{P}(a \in A) = p$, where $p = N^{-2/3 + \epsilon}$ for some fixed $\epsilon > 0$. My ...

**4**

votes

**0**answers

210 views

### The original proof of Szemerédi's Theorem

Nowadays there are plenty of different proofs of the celebrated Szemerédi's Theorem but for historical reasons I would like to read and understand the original proof. The proof is very tricky and, for ...

**4**

votes

**0**answers

50 views

### Are the extremal points of a certain set of functions $\mathcal P(\mathbf N) \to \bf R$ weakly additive?

Let an upper density (on $\mathbf N$) be a (set) function $f: \mathcal P(\mathbf N) \to \mathbf R$ such that, for all $X, Y \subseteq \bf N$ and $h,k \in \mathbf N^+$, the following hold:
(F1) ...

**3**

votes

**1**answer

144 views

### Freiman-isomorphic sets

Haw can we prove that an arbitrary set $A$ of $n$ positive integers is 2-Freiman isomorphic to a subset of {$ 1,2,...,4^{n}$} and $4^{n}$ cannot be improved to $2^{n}$?

**4**

votes

**1**answer

144 views

### Reference to a variant of Abel's summation formula

Edit. A stronger version of the formula is true (details follow).
Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, $(\lambda_n)_{n \ge 1}$ a nondecreasing sequence of positive reals such that ...

**2**

votes

**1**answer

54 views

### l-wise t-intersecting families of shifts of finite sets of integers

Let $A$ be a finite set of non-negative integers and write $I_k$ for the set ${0,1,\ldots,k-1}$. Form all possible l-wise intersections $(A+k_1)\cap \ldots \cap (A+k_l)$, where each $k_i$ runs through ...

**3**

votes

**2**answers

100 views

### Partition regular systems: do they have solution in (very dense) set of integers?

A partition regular system is a linear system of equations of the form $A\cdot x=0$, which satisfies a Ramsey-type result (namely, that for each $r>0$ whenever we colour the integers in $r$ ...

**3**

votes

**1**answer

214 views

### Some divisibility constraints in Frobenius coin problem

Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$.
Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ excellent if linear form ...

**2**

votes

**1**answer

89 views

### Intersections of translates of finite sets of integers

I am searching for a result in the literature that I am sure must be known, but I just fail to find it.
Let us starts with a simple example:
Let $A, B\subset \mathbb{Z}$ be a finite sets of integers ...

**4**

votes

**1**answer

251 views

### Problem related to Frobenius coin problem

Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$.
Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ good if,
for any ...

**1**

vote

**0**answers

74 views

### Consecutive integers divisible by consecutive small numbers

Given $n$, what is the largest set of consecutive integers in $[n,2n]$ can we have so that each integer is divisible by a distinct element from $[\log n,2\log n]$ (no partiular order)? So apriori I am ...

**5**

votes

**0**answers

65 views

### Some questions about the Lévy monoid of certain densities

Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$.
Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: ...

**0**

votes

**0**answers

48 views

### $\mathsf{GCD}$s of random linear form

Given $a,b\in\Bbb N_{<M}$ where $M\in\Bbb N_{>\exp(18)}$ is arbitrary with $(a,b)=1$, the probability that $\mathsf{gcd}(ax_1+by_1,ax_2+by_2)=1$ where $x_1,x_2,y_1,y_2\in\Bbb N_{>\ln M}$ is ...

**3**

votes

**1**answer

140 views

### Exact statistics in the Frobenius coin problem

The Frobenius coin problem guarantees that if $(a,b)=1$, then
$$ax+by$$ does not represent exactly $\frac{(a-1)(b-1)}2$ numbers all below $g(a,b)=ab-a-b$ if $x,y\geq0$ holds.
Assume $m\in[0,ab-a-b]$ ...

**8**

votes

**0**answers

175 views

### An abstract zero-sum problem

I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...

**12**

votes

**1**answer

744 views

### Where did the term “additive energy” originate?

A fundamental object in modern additive combinatorics and harmonic analysis is additive energy. Given a subset $A$ of (say) an abelian group $G$ the additive energy of $A$ is defined to be the ...

**7**

votes

**0**answers

251 views

### A characterization of quadratics similar to an inverse sieve problem

Suppose $\mathscr{A} \subset \mathbb{N}$ enjoys for all large enough cutoffs $X$ the following properties:
$|\mathscr{A} \cap [1,X]| > \sqrt{X}/10$; and
the discriminant $\prod_{\alpha \neq ...

**8**

votes

**0**answers

161 views

### Partition regularity in the squares

A linear equation $c_1x_1 + \cdots + c_sx_s = 0$ is partition regular if for every partition of the natural numbers into colour classes $A_1, \ldots, A_r$, there is a solution to the equation in which ...

**5**

votes

**0**answers

176 views

### Is $AA+A$ always at least as large as $A/A$?

Let $A$ be a finite set of real numbers. Is it always the case that $|AA+A| \geq |A/A|$?
In the line above, $AA+A:=\{ab+c:a,b,c \in A \}$, while $A/A:=\{a/b:a,b \in A, b\neq 0 \}$ is the ratio set.
...

**6**

votes

**1**answer

244 views

### Subsets of [1..N] with no three-term arithmetic progressions and no large gaps

Let S be a subset of [1..N] containing no three-term arithmetic progression, and let h(S) be the size of the largest gap between two consecutive elements of S. By Roth's theorem, h(S) has to grow ...

**6**

votes

**2**answers

246 views

### Additive energy of Piatetski-Shapiro sequences

Let $c>1$, and let $A$ denote the set
$$
\Big\{ \lfloor n^c \rfloor, \quad 1 \leq n \leq N \Big\}.
$$
Thus $A$ consists of the first $N$ elements of a so-called Piatetski-Shapiro sequence.
The ...

**1**

vote

**1**answer

150 views

### limit and combinatorics

Given $x \in (0,\frac{1}{2})$ and $y \in (0,\frac{1}{2}]$, what is the value of the following limit:
$\lim_{n\rightarrow \infty}\sum_{k=0}^{n}{n \choose k}|x^{n-k}(1-x)^{k}-y^{n-k}(1-y)^{k}|?$
When ...

**2**

votes

**0**answers

111 views

### Additive combinatorics and a Diophantine equation

Let $(n_k)_{1 \leq k \leq N}$ be a sequence of distinct positive integers. For $v \in \mathbb{Z}$ set
$$
A_N(v) = \# \Big\{ (k,\ell) \in \{1, \dots, N\}^2, ~k \neq \ell:\quad n_k - n_\ell = v \Big\}.
...

**13**

votes

**1**answer

488 views

### Erdös-Turán via Hardy-Littlewood circle method?

For any set $B\subseteq \mathbb{N}$ one can associate the formal series
$$f_B(z) = \sum_{b\in B}z^b$$
and obtain
$$f_B(z)^k = \sum_{n\geqslant 0} r_{B,k}(n)z^n,$$
where $r_{B,k}(n) = ...

**3**

votes

**2**answers

306 views

### Is the sumset or the sumset of the square set always large?

Let A be a finite subset of $\mathbb{N}$, $\mathbb{R}$, or a sufficiently small subset of $\mathbb{F}_{p}$.
Do we have a lower bound of the form $|A|^{1+\delta}$ on the following quantity:
$$\max ...

**6**

votes

**1**answer

304 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 ...

**4**

votes

**0**answers

207 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 ...

**18**

votes

**0**answers

485 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 ...

**6**

votes

**1**answer

266 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

**1**answer

77 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

**0**answers

102 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

**2**answers

279 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 ...

**2**

votes

**1**answer

196 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 ...

**9**

votes

**1**answer

424 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 of 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. Šalát's paper: ...

**7**

votes

**0**answers

153 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

**0**answers

130 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 ...

**3**

votes

**1**answer

179 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

**1**answer

180 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 ...

**5**

votes

**1**answer

167 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 ...

**2**

votes

**0**answers

66 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 ...

**4**

votes

**1**answer

250 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 $\theta(X) = 0$ ...

**2**

votes

**1**answer

111 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

**1**answer

123 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 ...

**3**

votes

**2**answers

371 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 ...

**4**

votes

**0**answers

99 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

**1**answer

298 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 ...

**1**

vote

**1**answer

203 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 ...

**10**

votes

**2**answers

406 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 ...