Former Exeter police officer Dan D'Amato got a man arrested in 2018 for criticizing him online after the man posted a comment on a local news website saying amont other things "This is the dirtiest most corrupt cop I have ever had the displeasure of knowing." He then began criticizng the police chief, William Shupe, for covering up for D'Amato. Thankfully the ACLU got involved and the state refused to pursue the charges.
This was a good example of police abusing their power to arrest people that they know could never be proven beyond a reasonable to have violate the law. All they need to do is claim to have probably cause that a specific law was broken. In the case of a law criminalizing false statements they need only claim to have probable cause that the statement is false. The problem with this is that a cop can abuse it simply by saying that he believes a statement of opinion to be false. The result is a chilling effect on speech that prevents people from speaking out against them because they know that even if they do not break the law that they can be punished by having to fight the charge. Fighting the charge requires that they be arrested, possibly detained pre-trial, pay for a lawyer, take time off work to go to court, etc.
The irony here is that I was unable to respond to the link to this story that a guy sent me on Facebook at the time because I was in jail for criticizing law enforcement in my area. In my case I was charged with a supervised release violation and then brutalized by jail guards that broke my arm. The guards accused me of assaulting them and I was prosecuted for assault resulting in bodily injury. I eventually pled to assault without bodily injury, but it was a selective prosecution. In my area similarly situated people are generally not charged federally for the same allegations. Those allegations being assault on a correctional officer by a federal inmate in a county jail resulting in minor injury in the form of fleeting pain. Of every similar case of such allegations my legal team found I was the only one prosecuted federally. The only difference was that I was speaking out against them.