Let $f\_n(x)\geq0$ be any sequence of nonnegative $L^1(\mathbb{R}^n)$ functions such that $\int\_{\mathbb{R}^{n}}f\_n(x)dx=1$ where $dx$ is the Lebesgue measure on $\mathbb{R}^n$. For any $a>1,\epsilon>0$, does there exist a sequence $x\_n\in\mathbb{R}^n$ such that
$$\lim\_{n\to+\infty}a^n\frac{\int\_{\|x-x\_n\|^2\leq n^{1-\epsilon}}f\_n(x)dx}{\int\_{\|x-x\_n\|^2\leq n}f\_n(x)dx}=0$$?


If it does not hold, is there any counter example of the sequence $f\_n$? What additional conditions do we need on the sequence $f\_n$? A similar question is to show that
$$\lim\_{n\to+\infty}a^n\frac{\int\_{\|x-x\_n\|^2\leq n^{1-\epsilon}}\exp(-\|x\_n-x\|^2)f\_n(x)dx}{\int\_{\|x-x\_n\|^2\leq n}\exp(-\|x\_n-x\|^2)f\_n(x)dx}=0$$
？This can be implied by the first display. Therefore, this is a weaker statement. 



Ratio of integrals with increasing dimension over Euclidean balls
Such a sequence $(x\_n)$ always exists. Indeed, the displayed ratio expression under the limit sign equals $R\_n(x\_n)$, where 
\begin{equation\*}
 R\_n(y):=\frac{g\_{n,r\_n}(y)}{g\_{n,s\_n}(y)},\quad 
 r\_n:=n^{(1-\epsilon)/2}, \quad 
 s\_n:=n^{1/2}, 
\end{equation\*}
\begin{equation\*}
 g\_{n,r}(y):=\int f\_n(x)1\_{|x-y|\_n<r}\,dx, 
\end{equation\*}
and $|\cdot|\_n$ is the Euclidean norm in $\mathbb R^n$. 
Note that for all real $r>0$
\begin{equation\*}
 \int g\_{n,r}(y)\, dy=\int dx\, f\_n(x)\int dy\,1\_{|x-y|\_n<r}=\int dx\, f\_n(x)|B\_n(r)|=|B\_n(r)|, 
\end{equation\*}
the volume of any ball in $\mathbb R^n$ of radius $r$, whence 
\begin{equation\*}
 \inf\_y \frac{g\_{n,r\_n}(y)}{g\_{n,s\_n}(y)}
 \le\frac{\int g\_{n,r\_n}(y)\, dy}{\int g\_{n,s\_n}(y)\, dy}=\Big(\frac{r\_n}{s\_n}\Big)^n=n^{-\epsilon n/2}. \tag{1}
\end{equation\*}
So, for some sequence $(x\_n)$ (depending only on $\epsilon$ but not on $a$) with $x\_n\in\mathbb R^n$ we have 
\begin{equation\*}
 R\_n(x\_n)=\frac{g\_{n,r\_n}(x\_n)}{g\_{n,s\_n}(x\_n)}<\inf\_y \frac{g\_{n,r\_n}(y)}{g\_{n,s\_n}(y)}+n^{-n}
 \le n^{-\epsilon n/2}+n^{-n}
=o(1/a^n) 
\end{equation\*}
for all real $a>0$, so that $a^nR\_n(x\_n)\to0$, 
as desired. 


**Details on the inequality in (1):** Let 
\begin{equation\*}
 c:=\inf\_y \frac{g\_{n,r\_n}(y)}{g\_{n,s\_n}(y)}. 
\end{equation\*}
Then $g\_{n,r\_n}(y)\ge c g\_{n,s\_n}(y)$ for all $y$, whence 
$\int g\_{n,r\_n}(y)\, dy\ge c\int g\_{n,s\_n}(y)\,dy$ and 
\begin{equation\*}
 \frac{\int g\_{n,r\_n}(y)\, dy}{\int g\_{n,s\_n}(y)\,dy}\ge c=\inf\_y \frac{g\_{n,r\_n}(y)}{g\_{n,s\_n}(y)}. 
\end{equation\*}


