Suppose that we have a Lazy 1-dimensional random walk $X_n$ valued in $\mathbb{Z}$, i.e. $$X_n = \sum_{i}^{n} \xi_i\;\;\;\;\;\;\;\;(\xi_i\;\text{iid}) $$ and $$\frac{1}{4}=P(\xi_1= 1)=P(\xi_1 =-1) $$$$\frac{1}{2}=P(\xi_1 = 0).$$
My question is, If $X_n$ starts at $0$, is it true that for any $|m| \leq \sqrt{n}$, we can have $$\exists c>0,s.t. P(X_n = m) \geq \frac{c}{\sqrt{n}},\;\text{ or }\; O(1/\sqrt{n})$$
I am trying to use the inversion formula and it boils down to integrate $\int_{-\pi}^{\pi} \cos^{2n}(\theta/2) \cos(m\theta)d\theta$ which seems don't result in a neat formula.
Any suggestions or ideas?