Let us say that any number which can be represented as a string of bits with a single decimal point is "binary enodable"; all such numbers are positive, computable, and rational.
Suppose we have some arbitrary set of integers, N. (ex. [16 32 40 48] )
All elements in N are binary encodable, with the important feature that the decimal point always has 0's to the left of it.
(ex. [1000.0 10000.0 101000.0 110000.0] )
Every element in N has a bit string representation; that bit strings length (in terms of bits needed to uniquely identify it) is M, and the summation of all M's in N, is K. (ex [21])
Now let's suppose we performed a map transform on N, in which we multiplied each element in N, by the same number C which is binary encodable, represented in the same format as the elements in N (with the crucial change that it may have bits to the left of the decimal), and stored the results of this transform into set T.
(ex. [1000.0 10000.0 101000.0 110000.0] * 0.001 = [10.0 100.0 101.0 110.0] )
T would then have it's own summation of the bit string representations of it's constituents (now possibly rationals rather than integers), which we will call S.
Here is my question: is there an effective procedure that will determine, for any set N, the computable real C which yields a set T with the smallest possible S?
Said more plainly, for any arbitrary set of integers is there any way to determine a rational factor which will yield the smallest representative numbers that still maintain the same scale of the original set?
