When your avenue is rank 1, you'll only get like 3 visitors a day.
When you get to around rank 10, that number will be more like 6. At rank 15, you'll have around 8 daily visitors to your avenue. So basically, the higher your rank, the more daily visitors your avenue gets.
Allow me to cast doubt on the accuracy of this...
When you use Fly to get to Join Avenue, you see the statistics of "Current visitors" and "People recommended/invited" below the map. I just reached rank 10, yet I still only get three visitors to my avenue per day.
However, when I last flew to the Avenue, I noticed that the "People recommended/invited" counter was at 96. I think it is this counter, and not the Avenue's rank, that brings in more people. Yes, it is closely tied to the rank of the avenue, and 100 visitors will often coincide very well with rank 10, but I haven't noticed any change in the number of visitors even when reaching that rank.
I guess I'll see on Thursday, when I finally reach 100 visitors (102, actually). If I get more people in then or on Friday, I'll take it as confirmation of my theory. If not, then it's bonkers.