Let be an irrational number, define , and let be the sequence obtained by arranging the elements of in increasing order. A sequence is said to be a signature sequence if there exists a positive irrational number such that , and is called the signature of .
One can also define two extended signature sequences for positive rational by taking the in increasing order or decreasing order. These can be considered signature sequences for and , respectively, where is an infinitesimal.
The signature of an irrational number or either signature of a rational number is a fractal sequence. Also, if is a signature or extended signature sequence, then the lower-trimmed subsequence is . It has been conjectured that every sequence with both of these properties is a signature or extended signature sequence.
If every initial subsequence of a sequence is an initial subsequence of some signature sequence, then is either a signature sequence, an extended signature sequence, or one of the two limiting cases: all 1's, or the natural numbers (which could be regarded as signature sequences for zero and infinity).