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Mordechay B. Levin has constructed a number $\lambda$ which is normal in base2, and such that the sequence $(\left\{2^n \lambda\right\})_{n=0,1,2,\ldots}$has very small discrepancy $D_N$. Indeed we have $N\cdot D_N = \mathcal{O}\left(\left(\log N\right)^2\right)$. This construction technique of Levin wasgeneralized by Becher and Carton, who generated normal numbers via perfectnested necklaces, and they showed that for these normal numbers the same upperdiscrepancy estimate holds as for the special example of Levin. In this papernow we derive an upper discrepancy bound for so-called semi-perfect nestednecklaces and show that for the Levin's normal number in arbitrary prime base$p$ this upper bound for the discrepancy is best possible, i.e., $N\cdot D_N\geq c\left(\log N\right)^2$ with $c>0$ for infinitely many $N$. This resultgeneralizes a previous result where we ensured for the special example of Levinfor the base $p=2$, that $N\cdot D_N =O( \left(\log N\right)^2)$ is bestpossible in $N$. So far it is known by a celebrated result of Schmidt that forany sequence in $[0,1)$, $N\cdot D_N\geq c \log N$ with $c>0$ for infinitelymany $N$. So there is a gap of a $\log N$ factor in the question, what is thebest order for the discrepancy in $N$ that can be achieved for a normal number.Our result for Levin's normal number in any prime base on the one hand mightsupport the guess that $O( \left(\log N\right)^2)$ is the best order in $N$that can be achieved by a normal number, while generalizing the class of knownnormal numbers by introducing e.g. semi-perfect necklaces on the other handmight help for the search of normal numbers that satisfy smaller discrepancybounds in $N$ than $N\cdot D_N=O( \left(\log N\right)^2)$.

Discrepancy bounds for normal numbers generated by necklaces in arbitrary base