Suppose that $X\sim \text{Bin}(n,\theta)$. Note that $X$ is the sum of $n$ $iid$ Bernoulli($\theta$) random variables. By the local limit theorem ([Theorem 7](https://terrytao.wordpress.com/2015/11/19/275a-notes-5-variants-of-the-central-limit-theorem/#more-8566) here) for the sum of discrete random variables, 
$$
P(X=t)=\frac{1}{\sqrt{2\pi n\theta(1-\theta)}}\exp\left(-\frac{(t-n\theta)^2}{2n\theta(1-\theta)} \right)+o(n^{-1/2})
$$ 
for all $n\geq 1$ and uniformly in the integers $t$.


Suppose $t=n\theta+\sqrt{2\theta(1-\theta)n\log m+O(1)}$. I am interested in the relationship between $n$ and $m$, where we can assume, $m>>n>>1$. For example, in my application, $m\approx 20000$ and $n\approx 200$ seems to work well. Intuitively, $m$ should grow much faster than $n$.


I'm interested in finding a theoretical relationship between $n$ and $m$ such that quantity $mP(X=t)=O(1)$. Now if the $o(n^{-1/2})$ remainder were not there, then I can reason the following,


\begin{align\*}
mP(X=t)&=O(mn^{-1/2}\exp(-O(\log m)))\\
&=O\left(\exp\left(\log m-\log n^{1/2}-O(\log m)\right)\right)\\
&= O\left(\exp\left(\log m^{r}-\log n^{1/2}\right)\right)\text{ for some constant $r>0$}\\
&= O\left(\frac{m^{r}}{n^{1/2}}\right)
\end{align\*}


This suggests that $m\leq n^\gamma$, where $\gamma = \frac{1}{2r}>0$ for $r>0$ can be a reasonable relationship. 


How do I handle that remainder $o(n^{-1/2})$? 



Deriving condition to get correct asymptotic bound
To get what you want, you can use the refinement of the local central limit theorem due to [Esseen, Theorem 5, page 63](https://projecteuclid.org/euclid.acta/1485888404), which in your case yields 
\begin{align}
 P(X=t)&=\frac1{\sqrt{npq}}\phi(x)\Big(1+\frac1{\sqrt n}\,Q\_k(x,1/\sqrt n)\Big)+o(1/n^{(k-1)/2}) %\\ 
 \tag{1}
\end{align}
as $n\to\infty$ uniformly in integer $t$, where $p:=\theta\in(0,1)$, $q:=1-p$, $\phi(x):=\frac1{\sqrt{2\pi}}\,e^{-x^2/2}$, 
\begin{equation}
 x:=\frac{t-np}{\sqrt{npq}},
\end{equation}
$k$ is any integer $\ge3$, and $Q\_k(x,1/\sqrt n)$ is a polynomial in $x,1/\sqrt n$. 


Suppose now that $m=n^c$, where $c=c(n)$ varies with $n$ so that $c\to c\_0$ for some real $c\_0>0$. Then for your $t=np+\sqrt{2pqn\ln m+O(1)}$ and $x$ as above we have 
\begin{equation}
 x=\sqrt{2\ln m+O(1/n)}=\sqrt{2c\ln n+o(1)}=o(n^a) \tag{2}
\end{equation}
for any real $a>0$, whence $\frac1{\sqrt n}\,Q\_k(x,1/\sqrt n)=o(1)$ in (1). 


Using the first equality in (2), we also have  

\begin{equation}
 \phi(x)\le e^{-x^2/2}\le\frac{1+O(1/n)}m=\frac{1+o(1)}m. 
\end{equation}
Using now (1) with $k>2c\_0+2$ (so that $n^{(k-1)/2}\ge n^{c+1/2}=m\sqrt n$ for all large enough $n$), we have  

\begin{equation}
 mP(X=t)=\frac m{\sqrt{npq}}\phi(x)(1+o(1))+o(m/n^{(k-1)/2})
 =O\Big(\frac1{\sqrt{n}}\Big)+o\Big(\frac1{\sqrt{n}}\Big)=O\Big(\frac1{\sqrt{n}}\Big),
\end{equation}
which is better than the $O(1)$ that you desired. 


