**35**

votes

**3**answers

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 $o(n^2)$?

**30**

votes

**4**answers

2k views

### Cliques, Paley graphs and quadratic residues

A question I've thought about, on and off for a long time, is how to improve the best bounds that (seem to be) known for the clique numbers of Paley graphs.
If p=1 mod 4 is a prime, we can define the ...

**5**

votes

**1**answer

481 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 ...

**20**

votes

**4**answers

2k views

### Number of vectors so that no two subset sums are equal

Consider all $10$-tuple vectors each element of which is either $1$ or $0$. It is very easy to select a set $v_1,\dots,v_{10}= S$ of $10$ such vectors so that no two distinct subsets of vectors $S_1 ...

**18**

votes

**0**answers

410 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 ...

**21**

votes

**3**answers

2k views

### How many different numbers can be obtained as product of first $n$ natural numbers?

Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set $\{1,2,\ldots,n\}$, that is consider the set
$\{1^{a_1} \cdot ...

**4**

votes

**2**answers

238 views

### Anticoncentration of the convolution of two characteristic functions

Edit: This is a question related to my other post, stated in a much more concrete way I think.
I am interested in anything (ideas, references) related to the following problem:
Suppose that $A ...

**3**

votes

**0**answers

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 ...

**3**

votes

**1**answer

159 views

### Is every sufficiently dense well mixed set an additive basis?

Let $B \subset \mathbb{N}$ be a set of natural numbers such that $|B \cap [1,N]| \sim N^\gamma$, for some $\gamma > 0$ with the following property:
For any pair of positive integers $k,n$ we ...

**85**

votes

**7**answers

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 ...

**24**

votes

**3**answers

1k views

### long enough interval of integers to solve a simultaneous congruence

Let $a$, $b$ be two coprime natural numbers. Let $A \subseteq \{0,1,\ldots, a-1\}$ and $B \subseteq \{0,1,\ldots,b-1\}$ be two nonempty sets, which we think of as sets of residues mod $a$ and $b$ ...

**20**

votes

**1**answer

2k views

### Monochromatic triangles in every two-coloring of the plane?

An old problem (possibly due to Erdős and Graham?): given a triangle $T$ and a two-coloring of the plane, does there necessary exist a monochromatic congruent copy of $T$? Here "monochromatic" means ...

**7**

votes

**2**answers

1k views

### Recent results on the Gauss circle problem?

Possible Duplicate:
What is the status of the Gauss Circle Problem?
The Gauss circle problem is the following: Let $N(r)$ denote the number of solutions in integer pairs $(i,j)$ to the ...

**6**

votes

**1**answer

359 views

### Upper bound for size of subsets of a finite group that contains a sum-full set

Problem
I'm looking for an upper bound for the number $k(G)$ of a finite group $G$, defined as follow:
Let $\mathcal{F}_k$ be the family of subsets of $G$ with size $k$, and we
define $k(G)$ be ...

**20**

votes

**3**answers

636 views

### A sumset inequality

A friend asked me the following problem:
Is it true that for every $X\subset A\subset \mathbb{Z}$, where $A$ is finite and $X$ is non-empty, that $$\frac{|A+X|}{|X|}\geq \frac{|A+A|}{|A|}?$$
...

**18**

votes

**0**answers

448 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 ...

**14**

votes

**3**answers

890 views

### The sum of integers being a bijection

What are the pairs $(P,Q)$ of subsets of $\mathbb N$ for which the map
\begin{eqnarray*}
P\times Q & \rightarrow & {\mathbb N} \\\\
(p,q) & \mapsto & p+q
\end{eqnarray*}
is a bijection ...

**12**

votes

**2**answers

790 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 ...

**7**

votes

**1**answer

284 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

**0**answers

144 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 ...

**6**

votes

**0**answers

331 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 ...

**4**

votes

**1**answer

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 ...

**4**

votes

**1**answer

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

**1**

vote

**1**answer

288 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 ...

**6**

votes

**2**answers

265 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 ...

**4**

votes

**1**answer

178 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 ...

**2**

votes

**2**answers

174 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 ...

**1**

vote

**1**answer

104 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 ...

**1**

vote

**1**answer

119 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 ...

**1**

vote

**1**answer

265 views

### Is $x + y \ne y+nx$ for $x \ne 0$ and $n \ge 2$ (in an ordered group)?

Let $(A, +, \preceq)$ be an ordered group, namely $(A, +)$ is a group and $\preceq$ is a total order on $A$ such that $x + z \prec y + z$ and $z + x \prec z + y$ for all $x,y,z \in A$ with $x \prec ...

**0**

votes

**0**answers

93 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 ...