+/- Beta-expansion is the use of non-integer numbers as the radix for number systems.
Certain algebraic Integers such as the quadratic Pisot numbers can be used as the radix of a number system to fully represent Q, by this I mean finitely or periodically like how the integers do. I don't know about all cubic Pisots but some can also fully represent Q such as the smallest Pisot number (the plastic constant). I was wondering if there were other Pisot and non-Pisot algebraics (such as sqrt2) that hold this finiteness property and moreso Is there a well defined set of numbers holding the property of full radix representation?
Some examples of finite representability are below:
2 = 100.00001(base 1.3247..) Radix is +solution to x^3 - x - 1 = 0.
2 = 10.01(base 1.61803..) Radix is +solution to x^2 - x - 1 = 0.
2 = 111.1(base -1.8393..) Radix is -solution to x^3 - x^2 - x - 1 = 0.
2 = 100(base 1.4142..) Radix is +solution to x^2 - 2 = 0.