Add to ORKGLet $\alpha=0.a_1a_2a_3\ldots$ be an irrational number in base $b>1$, where $0\leq a_i<b$. The number $\alpha \in (0,1)$ is a \textit{normal number} if every block $(a_{n+1}a_{n+2}\ldots a_{n+k})$ of $k$ digits occurs with probability $1/b^k$. A proof of the normality of the real number $\sqrt{2}$ in base $10$ is presented in this note. Three different proofs based on different methods are given: a conditional proof, and two unconditional proofs.Comment: Eighteen Pages. Keywords: Irrational number; Normal number; Uniform distribution; Borel proble
Given an integer q>1, a q-normal number is an irrational number r such that any preassigned sequence...
AbstractWe use entropy rates and Schur concavity to prove that, for every integer k⩾2, every nonzero...
This paper is centered around proving the irrationality of some common known real numbers. For sever...
In 1914, Felix Hausdorff published an elegant proof that almost all numbers are simply normal in bas...
A number is normal to the base r if, in its expansion to that base, all possible digit strings of le...
The first estimate of the upper bound $\mu(\pi)\leq42$ of the irrationality measure of the number $\...
It is well known that almost all real numbers (in the sense of Lebesgue measure) are normal to base ...
We use probabilistic methods, along with other techniques, to address three topics in number theory ...
For an integer b ≥ 2 a real number α is b -normal if, for all m > 0, every m-long string of digits i...
Let b ≥ 2 be an integer. A real number is called simply normal to base b if in its representation to...
We consider the real number σ with continued fraction expansion [a0, a1, a2,...] = [1, 2, 1, 4, 1, ...
Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its b...
summary:Let $Q=(q_n)_{n=1}^\infty $ be a sequence of bases with $q_i\ge 2$. In the case when the $q_...
Abstract. Let Bn(x) denote the number of 1’s occuring in the binary expansion of an irrational numbe...
AbstractWe show that the number generated by the q-ary integer part of an entire function of logarit...
Given an integer q>1, a q-normal number is an irrational number r such that any preassigned sequence...
AbstractWe use entropy rates and Schur concavity to prove that, for every integer k⩾2, every nonzero...
This paper is centered around proving the irrationality of some common known real numbers. For sever...
In 1914, Felix Hausdorff published an elegant proof that almost all numbers are simply normal in bas...
A number is normal to the base r if, in its expansion to that base, all possible digit strings of le...
The first estimate of the upper bound $\mu(\pi)\leq42$ of the irrationality measure of the number $\...
It is well known that almost all real numbers (in the sense of Lebesgue measure) are normal to base ...
We use probabilistic methods, along with other techniques, to address three topics in number theory ...
For an integer b ≥ 2 a real number α is b -normal if, for all m > 0, every m-long string of digits i...
Let b ≥ 2 be an integer. A real number is called simply normal to base b if in its representation to...
We consider the real number σ with continued fraction expansion [a0, a1, a2,...] = [1, 2, 1, 4, 1, ...
Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its b...
summary:Let $Q=(q_n)_{n=1}^\infty $ be a sequence of bases with $q_i\ge 2$. In the case when the $q_...
Abstract. Let Bn(x) denote the number of 1’s occuring in the binary expansion of an irrational numbe...
AbstractWe show that the number generated by the q-ary integer part of an entire function of logarit...
Given an integer q>1, a q-normal number is an irrational number r such that any preassigned sequence...
AbstractWe use entropy rates and Schur concavity to prove that, for every integer k⩾2, every nonzero...
This paper is centered around proving the irrationality of some common known real numbers. For sever...