# Thread: Problem on Residual Sets

1. ## Problem on Residual Sets

Cheers everyone. I'm going through a paper by Paul Erdos and am slightly flummoxed by one of the steps in his proof. Here goes:

Let $\displaystyle A$ and $\displaystyle B$ be residual sets in $\displaystyle \mathbb{R}$ (in the sense of Baire category). For a fixed real number $\displaystyle x$, the sets $\displaystyle A_x = \{x - a : a \in A\}$ and $\displaystyle B_x = \{x / b : b \in B\}$ are also residual.

I can see why this is true intuitively, particularly for $\displaystyle A_x$, since a simple linear transformation of the elements of $\displaystyle A$ isn't going to effect the category of the complementary set, but I can't seem to drum up a rigorous proof. No doubt it is going to involve the nowhere density of related sets, but I can't seem to connect the dots. Any help would be greatly appreciated. Many thanks.

2. Sure you mean $\displaystyle A_x$ is necessarily residual?

3. Originally Posted by Rebesques
Sure you mean $\displaystyle A_x$ is necessarily residual?
That is what I'm trying to prove; $\displaystyle A$ residual in $\displaystyle \mathbb{R} \implies A_x$ residual in $\displaystyle \mathbb{R},$ and $\displaystyle B$ residual in $\displaystyle \mathbb{R} \implies B_x$ residual in $\displaystyle \mathbb{R}.$

Erdos states this without proof in a paper of his, and I am trying to understand why it is true.