"Maybe the rich can save more because thy earn more."That's certainly a big part of it, but far from the only one.
When you're living paycheck to paycheck you have no money to invest, so the poor are usually limited to earning only as much as their poorly paying jobs will pay them, while the salaries of the rich usually only make up a small to non-existent portion of their wealth.
On top of that, at least in the US, capital gains are taxed at a much lower rate than income, so those whose income makes up a large fraction of what they earn are doubly screwed.
For the really poor, food expenses are a significant portion of their expenses, and yet many of them live in food deserts where food is not only less nutritious and less varied than what the rich have access to, but that food is often more expensive.
If as a poor person you have to take out a payday loan to get by, you're also going to be paying through the nose for that.
Predatory lending practices are also widely employed against the poor.
If you can't afford your own home (or to own a home but never pay it off), you could wind up paying way more in rent or mortgage payments than if you had bought a home outright in cash and never had to pay any interest on it at all.
That's not to mention medical expenses, which often bankrupt people in America... or legal expenses, which the poor often can't afford and wind up going to jail because of.
Finally, its the wealthy and powerful who write the laws and and use their connections in and various forms of bribes of government to favor mostly themselves, their friends and families. The poor do not have this sort of pull, and are often led by the nose by the rich to vote against their own interests.