The most famous in recent memory is the 2014 Study "Testing Theories of American Politics: Elites, Interest Groups, and Average Citizens" (https://doi.org/10.1017/S1537592714001595)
Which Concludes
"In the United States, indicate, the majority does not rule at least not in the causal sense of actually determining policy outcomes. When a majority of citizens disagrees with economic elites or with organized interests, they generally lose ... majorities of the American public actually have little influence over the policies our government adopts."
The totals for how many voters expressed which opinions are what carries the most sway with them. Whether or not those tallies are in alignment with the general public is not relevant.
So, when you contact them, you'll get a canned response telling you what their current stance is, regardless of whether that's the same as yours. But make no mistake, your opinion was added to the count.
This sort of thing is why phone banks and contact-your-representative campaigns are so common. In bulk, they're effective and can change policy even in ways that most of their constituents don't agree with. They're only looking at the numbers of contacts they've received.
