$G$ is a locally compact Hausdorff topological group, $A$ and $B$ are two Borel subsets of $G$, and $A$ and $B$ are both of the second category in $G$, then there exist an element $x\in G$, such that $A\bigcap xB$ is of the second category, too.
This problem is too difficult for me. I think for a long time but still cannot solve it out. Help me please.
Thanks in advance.
It seems the following.
We again should use the Baire Property. I recall that a subset $B$ of a topological space $X$ has the Baire Property in $X$ if $B$ contains a $G_\delta$-subset $C$ of $X$ such that $B\setminus C$ is meager in $X$. By [Kech, 8.22] each Borel subset of a space $X$ has the Baire Property in $X$. Now we use the following theorem from [Kech].
Since $A$ and $B$ are both of the second category in $G$, both sets $U(A)$ and $U(B)$ are non-empty. Pick arbitrary elements $a\in U(A)$ and $b\in U(B)$. Put $x=ab^{-1}$. Then a set $C=U(A)\cap xU(B)\ni a$ is an open non-empty set of a locally compact Hausdorff space (which is Baire, for instance, by [Kech, 8.4]). Therefore the set $C$ is of the second category. Then the set $A\cap xB$ is of the second category too.
References
[Kech] A. Kechris. Classical Descriptive Set Theory, – Springer, 1995.