Let $A$ be a finite set of real (or complex) numbers. If I consider sets with small sizes, we have that: If $A$ is the empty set, then $A+A$ is also empty. If $A$ is a singleton, then $A+A$ is also a singleton. If $|A|=2,$ then $|A+A|=3.$ If $|A|=3,$ then $|A+A|$ can be at most […]

I am going to ask here a generalization of this other question: Problem Fix $\varepsilon>0$ and let $p$ be a sufficiently large prime. Then, show that, for every $X\subseteq \{1,\ldots,p\}$ with $|X|\ge \varepsilon p$, there exist $a,b,c,x,y,z \in X$ such that $p$ divides $ax+by+cz$. Motivation: I know a solution for this problem, but I feel […]

The Minkowski sum of closed sets needn’t be closed; $\mathbb{Z} + \sqrt{2}\mathbb{Z}$ is the canonical example. However, its not clear to me how to prove this. Question. How can we prove that $\mathbb{Z} + \sqrt{2}\mathbb{Z}$ isn’t closed?

The Sum of 2 sets of Measure zero might well be very large for example, the sum of $x$-axis and $y$-axis, is nothing but the whole plane. Similarly one can ask this question about Cantor sets: If $C$ is the cantor set, then what is the measure of $C+C$?

Why is it that if we have a compact set $X$ and a closed set $Y$ then the Minkowski sum $X+Y$ is necessarily closed? (Sorry for keep asking questions about the Minkowski sum, I am trying to figure out how these things work.) Thanks.

I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin’s Functional Analysis), where $A$ and $B$ are subsets of a topological vector space $X$. In case $X=\mathbb{R}$ this is easy, using sequences. However, since I was told that using sequences in topology is […]

I am trying to prove that $C+C =[0,2]$ ,where $C$ is the Cantor set. My attempt: If $x\in C,$ then $x= \sum_{n=1}^{\infty}\frac{a_n}{3^n}$ where $a_n=0,2$ so any element of $C+C $ is of the form $$\sum_{n=1}^{\infty}\frac{a_n}{3^n} +\sum_{n=1}^{\infty}\frac{b_n}{3^n}= \sum_{n=1}^{\infty}\frac{a_n+b_n}{3^n}=2\sum_{n=1}^{\infty}\frac{(a_n+b_n)/2}{3^n}=2\sum_{n=1}^{\infty}\frac{x_n}{3^n}$$ where $x_n=0,1,2, \ \forall n\geq 1$. Is this correct?

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