Normality of Irrational Algebraic Numbers

Normality of Irrational Algebraic Numbers

Definitions and mathematical statement

Write the fractional part of a real number in base $b\ge2$ as $0.d_1d_2\ldots$. It is normal to base $b$ if for every length $k$ and every word $w\in\{0,\ldots,b-1\}^k$, \[ \lim_{N\to\infty}\frac{\#\{1\le j\le N:(d_j,\ldots,d_{j+k-1})=w\}}{N}=b^{-k}. \] The conjecture requires this for every real algebraic irrational number and every integer base. Algebraic means a root of a nonzero rational polynomial. Frequency of individual digits alone is weaker than normality of all blocks.

Formulation and concept references: [S1].

Short English statement

Do the digits of every algebraic irrational have the uniform block frequencies of a normal number in every integer base?

Research attempt

Current frontier and scope.

The target is normality of every real algebraic irrational in every integer base, including all finite digit blocks. The source discusses the distance between known transcendence and digit-structure theorems and this frequency assertion. [S1] A relevant established result is Adamczewski--Bugeaud's theorem that the block complexity $p(\ell)$ of an algebraic irrational's base-$b$ expansion satisfies $p(\ell)/\ell\to\infty$. This counts distinct blocks and is not a statement about their frequencies. [E1] The search through 27 September 2026 did not provide a verified theorem proving the full normality statement. The older structural theorem is used below as a benchmark, not represented as a complete current inventory of digit estimates.

Attack map.

A Diophantine-approximation route tries to turn a biased digit expansion into excessively good rational approximations. Long constant runs can indeed be bounded, but frequency bias need not create them. An exponential-sum route uses algebraicity in a cancellation estimate for a lacunary sequence; after differencing, the same difficult type of sum reappears. A combinatorial route would strengthen the known block-complexity theorem into uniform frequencies, but maximal block complexity by itself is insufficient. These routes are compared using explicit deductions and counterexamples to their proposed intermediate inferences.

Attempt I: the exact exponential-sum target.

Replace $\alpha$ by its fractional part, which remains an algebraic irrational in $(0,1)$. Fix a base $b\geq2$. For $x_n=\{b^n\alpha\}$, a word of length $\ell$ representing the integer $j\in\{0,\ldots,b^\ell-1\}$ starts at digit $n+1$ exactly when \[ x_n\in[j/b^\ell,(j+1)/b^\ell). \] No $x_n$ is an endpoint, since that would make $\alpha$ rational. Consequently equidistribution of $(x_n)$ implies normality. Conversely, if all these block frequencies are correct, approximate any interval from inside and outside by unions of base-$b$ intervals of mesh $b^{-\ell}$. Their total length difference tends to zero, giving uniform distribution in every interval. Thus the two assertions are equivalent, with no density-one subsequence substituted.

The corresponding Fourier criterion is \begin{equation}\label{wsref-0-31-32-35-39-3a-77-65-79-6c} \forall h\in{\mathbb{Z}}\setminus\{0\}:\qquad S_h(N):=\sum_{n=0}^{N-1}\exp(2\pi i h b^n\alpha)=o(N). \end{equation} To recall why, equidistribution gives the averages of each nonconstant Fourier character equal to its integral zero. Conversely, Fourier averages determine averages of trigonometric polynomials; uniform approximation of continuous periodic functions by such polynomials, followed by continuous upper and lower approximations to interval indicators, gives equidistribution. These are exact logical reductions, not a new cancellation theorem. The source's discussion of uniform digit distribution provides the original context. [S1]

Attempt II: differencing and its circular stopping point.

Put $z_n=\exp(2\pi i h b^n\alpha)$ for $0\leq n<N$ and extend it by zero outside that range. For $1\leq H\leq N$, summing overlapping blocks and applying Cauchy--Schwarz gives \begin{equation}\label{wsref-0-31-32-35-39-3a-76-64-63} |S_h(N)|^2\leq\frac{N+H-1}{H^2} \left(HN+2\sum_{d=1}^{H-1}(H-d)\Re C_d(N)\right), \end{equation} where \[ C_d(N)=\sum_{n=0}^{N-d-1}z_{n+d}\overline{z_n} =\sum_{n=0}^{N-d-1} \exp(2\pi i h(b^d-1)b^n\alpha). \] For the inequality, use $HS_h(N)=\sum_{r=-(H-1)}^{N-1}\sum_{j=0}^{H-1}z_{r+j}$. Expanding the squared inner sums counts a pair separated by $d$ exactly $H-d$ times, which proves (1259:vdc) including its endpoints and constant factors.

For any fixed $H$, bounds $C_d(N)=o(N)$ for $1\leq d<H$ would imply $\limsup_N|S_h(N)|^2/N^2\leq1/H$; then $H\to\infty$ would prove (1259:weyl). But $C_d(N)$ is precisely another sum of the same form, with frequency $h(b^d-1)$. Multiplying an algebraic irrational by this nonzero integer preserves algebraicity and irrationality; it supplies no quantitative cancellation. Invoking normality of that new algebraic number to bound the correlation would assume an instance of the very universal conjecture being attempted. The useful missing lemma is an independent bound for these correlations, not the formal differencing identity.

Attempt III: what Roth-type approximation actually forbids.

Suppose that after its first $m$ base-$b$ digits, $\alpha$ has $L$ consecutive zeros. If the prefix represents $p/b^m$, then \[ 0<|\alpha-p/b^m|\leq b^{-m-L}. \] For every fixed $\varepsilon>0$, Roth's approximation theorem implies a constant $c_{\alpha,\varepsilon}>0$ such that every reduced rational $p'/q$ satisfies $|\alpha-p'/q|\geq c_{\alpha,\varepsilon}q^{-2-\varepsilon}$. The finitely many exceptional approximants are incorporated by reducing the positive constant. This theorem and its digit applications are part of the source's Diophantine-approximation discussion. [S1]

Reducing $p/b^m$ gives $q\leq b^m$, so the lower bound is at least $c_{\alpha,\varepsilon}b^{-(2+\varepsilon)m}$. Combining the two inequalities yields \begin{equation}\label{wsref-0-31-32-35-39-3a-72-75-6e-73} L\leq(1+\varepsilon)m+O_{\alpha,b,\varepsilon}(1). \end{equation} A run of digits all equal to $b-1$ is treated with the upper rational endpoint $(p+1)/b^m$. Its difference from $\alpha$ is nonzero for an irrational number, and the same bound follows. No effective value for Roth's constant is asserted.

This proof rules out certain very long uniform runs. It does not turn a fixed positive digit-frequency bias into a contradiction: a biased sequence can change digits often enough to have no long runs at all. Nor does the block-complexity theorem $p(\ell)/\ell\to\infty$ provide an estimate for the relative frequencies required by normality. [E1]

A probabilistic stress test of both structural shortcuts.

Consider independent binary digits with probabilities $\Pr[d_n=1]=1/3$ and $\Pr[d_n=0]=2/3$. With probability one, the frequency of ones is $1/3$, by the strong law, so the resulting number is not normal to base two. Every fixed finite word has positive probability at each of infinitely many disjoint blocks. Independence shows that it appears infinitely often almost surely. Intersecting these probability-one events over the countably many words gives \[ p(\ell)=2^\ell\quad\text{for every }\ell \] almost surely: the block complexity is maximal, not merely superlinear.

The same sequences almost surely have only logarithmically long constant runs. Among the first $N$ digits, the probability of a constant run of length $L$ is at most $2N(2/3)^L$. Take $N=2^j$ and $L=\lceil C\log N\rceil$ with $C$ sufficiently large; the resulting probabilities sum to a finite number over $j$. The union bound and the elementary first Borel--Cantelli lemma then give a logarithmic bound for all large dyadic $N$, and hence for all large $N$. This is far stronger than (1259:runs) while still compatible with persistent frequency bias.

These numbers are not proposed algebraic counterexamples. The construction demonstrates that the two consequences extracted above---many distinct blocks and no excessively long constant runs---do not logically imply normality, even together. A successful proof must exploit further arithmetic information about algebraicity, not just these combinatorial consequences. Conversely, no theorem here says that every biased sequence satisfying these conditions is transcendental.

Exact finite digit experiments.

To check digit-frequency computations without floating-point ambiguity, let $N\geq1$ and use the integer \[ M=\left\lfloor\sqrt{2b^{2N}}\right\rfloor. \] Its defining certificate is the exact pair of inequalities $M^2\leq2b^{2N}<(M+1)^2$. The $N$ base-$b$ digits of $M-b^N$, padded with leading zeros, are exactly the first $N$ fractional digits of $\sqrt2$. The companion script applies this to a binary prefix of length $4096$ and a decimal prefix of length $1000$, and records all overlapping block counts for lengths one through three.

For these finite windows there are $N-\ell+1$ complete length-$\ell$ blocks, not $N$ blocks with a silently omitted suffix. The report states that denominator explicitly. The integer-square certificate proves that the tested digits are correct; it says nothing about the asymptotic behavior of their frequencies, or about another algebraic number or another base.

The binary prefix has $2066$ zeros and $2030$ ones. Its $4094$ complete overlapping triples have the following exact counts: \[ \begin{array}{c|r@{\qquad}c|r} \text{word}&\text{count}&\text{word}&\text{count}\\\hline 000&555&100&516\\ 001&516&101&477\\ 010&501&110&493\\ 011&493&111&543 \end{array} \] They sum to $4094$, not $4096$; the largest absolute deviation from frequency $1/8$ is exactly $173/16376$. The decimal prefix yields \[ \begin{array}{c|rrrrrrrrrr} \text{digit}&0&1&2&3&4&5&6&7&8&9\\\hline \text{count}&108&98&109&82&100&104&90&104&113&92 \end{array} \] with denominator $1000$. These outputs illustrate the difference between an exact finite-prefix certificate and the all-length limiting assertion: even correct counts for all words in this finite window do not bound the discrepancy at later positions. The full two- and three-digit decimal tables are retained in the machine-readable report.

Outcome.

Outcome: Exploratory approach with unresolved gap.

The Fourier equivalence, differencing identity, and long-run bound are proved. The probabilistic example refutes the proposed implication from maximal block complexity and short constant runs to normality. The arithmetic cancellation estimate (1259:weyl) remains unproved; no algebraic counterexample, resolution, or historical novelty is claimed.

Sources

• S1 Michel Waldschmidt, Words and Transcendence, 2005. https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/WordsTranscendence.pdf. Catalogue formulation source.
• E1 Boris Adamczewski and Yann Bugeaud. On the complexity of algebraic numbers I. Expansions in integer bases, Annals of Mathematics 165 $2007$, no. 2, 547--565, Theorem 1 (block complexity, not frequency normality). https://arxiv.org/html/math/0511674. Consulted 27 September 2026 for the indicated result or scope.