22 research outputs found
Discrepancy bounds for normal numbers generated by necklaces in arbitrary base
Mordechay B. Levin has constructed a number which is normal in base
2, and such that the sequence
has very small discrepancy . Indeed we have . This construction technique of Levin was
generalized by Becher and Carton, who generated normal numbers via perfect
nested necklaces, and they showed that for these normal numbers the same upper
discrepancy estimate holds as for the special example of Levin. In this paper
now we derive an upper discrepancy bound for so-called semi-perfect nested
necklaces and show that for the Levin's normal number in arbitrary prime base
this upper bound for the discrepancy is best possible, i.e., with for infinitely many . This result
generalizes a previous result where we ensured for the special example of Levin
for the base , that is best
possible in . So far it is known by a celebrated result of Schmidt that for
any sequence in , with for infinitely
many . So there is a gap of a factor in the question, what is the
best order for the discrepancy in that can be achieved for a normal number.
Our result for Levin's normal number in any prime base on the one hand might
support the guess that is the best order in
that can be achieved by a normal number, while generalizing the class of known
normal numbers by introducing e.g. semi-perfect necklaces on the other hand
might help for the search of normal numbers that satisfy smaller discrepancy
bounds in than .Comment: 29 page