Given $x_1,x_2,\ldots,x_n$ i.i.d. bernoulli random variables with $P(x_i=1)=\frac1n$. Given a constant $c>1$. Is there an explicit theorem that can derive a concentration argument (possibly a modified Chernoff or similar) for the following:

$$P(\sum_{i=1}^n c^{i-1} \cdot x_i - E(x_i)\cdot \sum_{i=1}^n c^{i-1}>\epsilon)<\delta$$

All the posts I found are for general weighted sum Bernoulli and there's not an explicit form yet, but here my question has more structure to it. Thanks!

Given $x_1,x_2,\ldots,x_n$ i.i.d. bernoulli random variables with $P(x_i=1)=\frac1n$. Given a constant $c=1+\frac{1}{m}, m\geq n$. Is there an explicit theorem that can derive a concentration argument (possibly a modified Chernoff or similar) for the following:

$$P(\sum_{i=1}^n c^{i-1} \cdot x_i - E(x_i)\cdot \sum_{i=1}^n c^{i-1}>\epsilon)<\delta$$

All the posts I found are for general weighted sum Bernoulli and there's not an explicit form yet, but here my question has more structure to it. Thanks!