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### Distortion of the Hausdorff dimension of sums of Cantor sets under local scaling

The following question deals with possible distortion of the Hausdorff dimension of sums of Cantor sets as one "zooms in" on the sum around any given point.
Let us assume that $C_1$ and $C_2$ are two ...

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### 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 ...

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### 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 ...

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193 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 ...

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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
$$
...

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### 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 ...

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### Number of orders of $k$-sums of $n$-numbers

Suppose we have a $n$-element set $S$. Denote the set of its $k$-element subsets by $K$ ($|K|=\binom{n}{k}$).
If the elements of $S$ are real numbers then to each $k$-element subset we can associate ...

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### Subgroup cliques in the Paley graph

It is a famous open problem to estimate non-trivially, for a prime $p\equiv 1\pmod 4$, the largest size of a subset $A\subset{\mathbb F}_p$ such that the difference of any two elements of $A$ is a ...

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341 views

### How to find an integer set, s.t. the sums of at most 3 elements are all distinct?

How to find a set $A \subset \mathbb{N}$ such that any sum of at most three Elements $a_i \in A$ is different if at least one element in the sum is different.
Example with $|A|=3$: Out of the set $A ...

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### $B_k[1]$ sets with smallest possible $m = max B_k[1]$ for given $k$ and $n = |B_k[1]|$ elements

Sidon sets are sets $A \subset \mathbb{N}$ such that for all $a_j,b_j \in A$ holds
$$a_1+a_2=b_1+b_2 \iff \{a_1,a_2\}=\{b_1,b_2\}$$
Thus if you know the sum of two elements, you know which elements ...

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### Sumsets and a bound

Let $q$ be a positive integer. Is it true there exists a constant $C_q$ such that the following inequality holds for any finite set $A$ of reals:
$$\displaystyle |A+qA|\ge (q+1)|A|-C_q\qquad (1)$$
I ...

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493 views

### Distribution of $a^2+\alpha b^2$

It is well known that size of the set of positive integers up to $n$ that can be written as $a^2+b^2$ is asymptotic to $C \frac{n}{\sqrt{\log n}}$. Here I'm interested mostly in the weaker fact that ...

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624 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 $$\frac{|A+X|}{|X|}\geq \frac{|A+A|}{|A|}?$$
...

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### 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 ...

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### Lower bounds for $|A+A|$ if $A$ contains only perfect squares

Let $A$ a set with $|A|=n$ that contains only perfect squares of integers.
What lower bounds can we give for $|A+A|$?
I think the lower bound $\gg \frac{n^2}{\sqrt{log \,n}}$ holds (this would be ...

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### 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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236 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 ...

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348 views

### A problem related with 'Postage stamp problem'

A friend of mine taught me this question. I found that it is related with 'Postage stamp problem' (though it does not seem to be same).
Let $m,a_1\lt a_2\lt \cdots\lt a_n$ be natural numbers. Now let ...

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473 views

### Size of a certain sumset in $\mathbb{Z}/p^2\mathbb{Z}$

We are interested in estimating the size of a certain sumset in $\mathbb{Z}/p^2\mathbb{Z}$. Let $p$ be an odd prime, $g$ a primitive root modulo $p^2$, and $A=\langle g^p\rangle$ the unit subgroup of ...

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### A sum-product estimate in Z/p^2Z

We are interested in a sum-product type estimate. Let $p$ be an odd prime, and let $A$ be the order $p-1$ subgroup of $(\mathbb{Z}/p^2\mathbb{Z})^\times$. That is, let $A = \langle g^p \rangle$, where ...

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### how to proof this Stirling related equation

here is what I need to proof, have no idea were to start. I know there is some connection with the Stirling theorem.
$$
\sum_{i=0}^{d}\binom{m}{i} \leq \left ( \frac{em}{d} \right )^{d}
$$
I tried ...

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**1**answer

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### unique sums in a finite direct product of sets of integers

I am an algebraist, and I am wondering if there is a definition for the following:
Let $A_1$, $A_2$, $\ldots, A_n$ be sets of integers (or more generally, subsets of a group $G$). Say that (for the ...

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344 views

### Additive set with small sum set and large difference set

I have a question!
Can someone explain how (the intuition, method?) one can try to construct an additive set of cardinality $N$ with a small sum set (around $N$) and a very large difference set ...

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### 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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176 views

### Find an approximate expression of a sum of a product using the average of each item

Is it possible to find an approximate expression of $\frac{\sum_{i=1}^{n} k_i x_i}{\sum_{i=1}^{n} k_i}$ using $\langle k \rangle$, $\langle k^2 \rangle$, $\langle x \rangle$, and $n$? Alternatively if ...

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### Intersecting Hamming spheres: is $|A\stackrel k+E|\ge|A|$?

Since my original posting some ten days ago, I discovered an amazing
example which changed significantly my perception of the problem.
Accordingly, the whole post got re-written now.
The most ...

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### The hypercube: $|A {\stackrel2+} E| \ge |A|$?

I have a good motivation to ask the question below, but since the post is
already a little long, and the problem looks rather natural and appealing
(well, to me, at least), I'd rather go straight to ...