Computing Absolutely Normal Numbers in Nearly Linear Time

@article{Lutz2016ComputingAN,
  title={Computing Absolutely Normal Numbers in Nearly Linear Time},
  author={Jack H. Lutz and Elvira Mayordomo},
  journal={ArXiv},
  year={2016},
  volume={abs/1611.05911}
}
A real number $x$ is absolutely normal if, for every base $b\ge 2$, every two equally long strings of digits appear with equal asymptotic frequency in the base-$b$ expansion of $x$. This paper presents an explicit algorithm that generates the binary expansion of an absolutely normal number $x$, with the $n$th bit of $x$ appearing after $n$polylog$(n)$ computation steps. This speed is achieved by simultaneously computing and diagonalizing against a martingale that incorporates Lempel-Ziv parsing… Expand
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References

SHOWING 1-10 OF 52 REFERENCES
On the construction of absolutely normal numbers
We give a construction of an absolutely normal real number $x$ such that for every integer $b $ greater than or equal to $2$, the discrepancy of the first $N$ terms of the sequence $(b^n x \modExpand
A polynomial-time algorithm for computing absolutely normal numbers
We give an algorithm to compute an absolutely normal number so that the first n digits in its binary expansion are obtained in time polynomial in n; in fact, just above quadratic. The algorithm usesExpand
Note on normal numbers
was normal (in the sense of Borel) with respect to the base 10, a normal number being one whose digits exhibit a complete randomness. More precisely a number is normal provided each of the digits 0,Expand
Turing's unpublished algorithm for normal numbers
In an unpublished manuscript, Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he gave an algorithm forExpand
Nearly Linear Time
TLDR
This work introduces a class NLT of functions computable in nearly linear time n(log n)O(1) on random access computers and gives also a machine-independent definition of NLT and a natural problem complete for NLT. Expand
Distribution Modulo One and Diophantine Approximation
1. Distribution modulo one 2. On the fractional parts of powers of real numbers 3. On the fractional parts of powers of algebraic numbers 4. Normal numbers 5. Further explicit constructions of normalExpand
Why Computational Complexity Requires Stricter Martingales
TLDR
This paper elucidates the relationship between these notions and proves that the latter notion is too weak for many purposes in computational complexity, b ecause under this definition every computable martingale can be simulated by a polynomial-time computableMartingale. Expand
Series Feasible Analysis , Randomness , and Base Invariance
We show that polynomial time randomness of a real number does not depend on the choice of a base for representing it. Our main tool is an ‘almost Lipschitz’ condition that we show for the cumulativeExpand
Feasible Analysis, Randomness, and Base Invariance
TLDR
It is shown that polynomial time randomness of a real number does not depend on the choice of a base for representing it, and it is proved that for any base r, n⋅log2n-randomness in base r implies normality in base  r, and that n4- randomness inbase r implies absolute normality. Expand
Computable Absolutely Pisot Normal Numbers
We analyze the convergence order of an algorithm producing the digits of an absolutely normal number. Furthermore, we introduce a stronger concept of absolute normality by allowing Pisot numbers asExpand
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