If you use a lookahead regex, you can avoid resetting the position after every match. But the real issue (in my mind) is that you use $&, which incurs a significant performance hit. Here's a solution that produces the same output using lookaheads and substr:

#! /usr/bin/perl -w

use strict;
use warnings;

my $a = "Input String: aaAabBAAa";

my $n = shift @ARGV;
$n --;

my $rx = qr/(.)(?=\1{$n})/i;
while ( $a =~ /$rx/g )
{
    print substr( $a, pos($a)-1, $n+1), "\n";
}
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