1 Background and Motivation

1.1 Numeral systems and digit expansions

Real numbers can be encoded in many numeral systems. Given a base \(b\ge 2\), each \(x\in[0,1)\) can be written (when not ending with a recurring tail of \(b-1\)’s) as an infinite digit sequence \[ x=0.d_1d_2d_3\ldots \quad (d_i\in\{0,1,\dots,b-1\}), \] which can also be viewed as a concatenation of blocks of digits. Many statistical properties of \(x\) are therefore properties of the long-run behavior of its digit blocks.

1.2 Normality versus non-normality

A “normal” number in base \(b\) is one whose digit blocks of any fixed length appear with the expected uniform frequencies. Non-normality arises when these frequencies fail to match the uniform baseline, either persistently or along infinitely many block lengths. Subnormality is a refinement that focuses on a particular type of non-uniformity: the deficit is not only present, but strong enough to violate certain growth or distribution benchmarks.

1.3 Subnormality as a spectrum of statistical behavior

Instead of a binary split into normal and non-normal, subnormality is best understood as part of a spectrum. Some numbers exhibit mild irregularities that disappear in the limit, while others show systematic biases or suppressed patterns. Subnormal numbers typically fall on the “more structured deviation” side: their digit statistics approach incorrect targets at a rate that is measurably slower than required by full normality.

1.4 Connections to uniform distribution concepts

Uniform distribution of sequences and digit blocks is closely tied to how evenly a sequence spreads across its state space. When digit blocks are treated as outputs of a long sequence, subnormality reflects a failure of uniform spread at the level of frequencies and discrepancies. This perspective aligns subnormal number questions with the broader toolkit of discrepancy theory and equidistribution.

2 Definitions and Core Concepts

2.1 Subnormal number in a base-b expansion

2.1.1 Digit-block frequency and limiting proportions

Fix a base \(b\ge 2\). Consider a block length \(k\) and a specific \(k\)-digit word \(w\in\{0,\dots,b-1\}^k\). Let \(N_w(n)\) be the number of occurrences of \(w\) among the first \(n\) digits (using the usual sliding-block convention, so there are \(n-k+1\) positions). If the digit frequencies were perfectly uniform, one expects \[ \frac{N_w(n)}{n} \approx b^{-k} \] for large \(n\). Subnormality is defined so that the deviation from the target proportions does not shrink as fast as it would for numbers meeting the “normal” criterion, or equivalently that some family of deviations violates a specified bound.

2.1.2 Sublinear growth conditions and exceptional sets

A common pattern in subnormality definitions is to demand that a certain discrepancy or deviation quantity grows at most on a “sublinear” scale, but then to consider numbers for which this requirement fails in a controlled way. One way to phrase this is: for a prescribed rate function \(r(n)\) that is smaller than linear growth, normal-like behavior would typically produce deviations bounded by something comparable to \(r(n)\); subnormal numbers are those whose deviations exceed the expected envelope along infinitely many \(n\), with the excess large enough to be statistically meaningful. The failure often occurs on carefully chosen “exceptional sets” of positions or along subsequences tied to the block structure.

2.2 Equivalent formulations

2.2.1 Relations to discrepancy and equidistribution

Discrepancy measures how far empirical frequencies differ from the uniform measure. Subnormality statements can be re-expressed as lower bounds on discrepancy for certain partitions induced by digit blocks. In this language, a number is subnormal when the sequence of its digit-block empirical measures does not approach the uniform measure quickly enough, or does so unevenly across block types.

2.2.2 Measure-theoretic viewpoints

From a measure-theoretic stance, one studies sets of numbers for which the digit statistics fall into “atypical” regimes. Subnormality then describes membership in sets of measure zero under the usual product measure that models independent uniform digits, but these sets can still be large in a category-theoretic sense (for example, dense or of full Hausdorff dimension, depending on the exact definition). This yields a precise framework for separating typical behavior (modeled by randomness assumptions) from structured anomalies.

2.3 Dependence on base and representation

Subnormality depends on the numeral base and on the convention used for expansions with ambiguous tails. Different bases induce different digit partitions and different collections of block words; a number can be subnormal in one base while exhibiting closer-to-normal behavior in another. Representation choices can matter for boundary cases, especially for rationals with terminating expansions; standard conventions typically exclude the ambiguity by using the non-terminating representation when needed.

3 Theoretical Properties

3.1 Existence and construction methods

Subnormal numbers exist in abundance because normality is a strong requirement. There are multiple construction approaches:

  • Digit-restriction constructions: enforce that certain words occur too rarely (or too often) by building expansions from a constrained language.
  • Sparsity and staging: define digits in stages whose later blocks are chosen to control frequency growth rates.
  • Algorithmic recipes: specify expansions via substitution rules or iterative pattern insertion that guarantees persistent deviation.

These methods produce explicit or semi-explicit examples and also help estimate how large the set of subnormal numbers can be under chosen definitions.

3.2 Typical versus exceptional behavior

3.2.1 Measure, category, and density heuristics

Under the heuristic model where digits behave like independent uniform random variables, normal-like frequency convergence is typical, and severe frequency distortions should have probability zero. Subnormal sets are therefore “rare” in measure. However, rarity in measure does not imply smallness in other senses: many exceptional sets of digit-constraint type can still be dense and large from the viewpoint of category or fractal dimension. The exact balance depends on how strong the subnormality criterion is.

3.3 Stability under transformations

3.3.1 Scaling and addition effects

Simple arithmetic operations can alter digit expansions in ways that complicate frequency properties. Multiplying by an integer base-related factor can introduce carries, shifting the digit pattern globally; adding a number may trigger long carry chains. As a result, subnormality is not universally stable under arithmetic transformations. When stability does hold, it typically requires either controlled carry behavior (e.g., for restricted ranges) or a definition formulated to be invariant under specific digit-level transformations.

3.3.2 Changing base and digit mapping

Changing bases generally changes the digit-block structure, so subnormality is usually not preserved without additional assumptions. Still, when bases are related (for example, one base is a power of another) and digit-grouping is compatible, one can sometimes translate subnormality statements through block regrouping. Similarly, digit mappings that permute digits within a base can preserve uniform frequency targets and thus may preserve subnormality for definitions tied to uniformity.

3.4 Cardinality and size of sets of subnormal numbers

For typical subnormality definitions, the set of subnormal numbers is uncountable. In many formulations it also has:

  • Lebesgue measure zero (with respect to the usual uniform digit product measure),
  • while still being large in fractal sense (positive Hausdorff dimension) or topologically large (dense or even residual in some contexts).

The exact size depends on the strength of the deviation condition, the chosen rate function, and which block lengths and patterns are tested.

4 Examples and Worked Illustrations

4.1 Simple digit-restricted constructions

4.1.1 Numbers with constrained block occurrences

A basic example begins by selecting a subset of digit words to underuse. For instance, fix a block length \(k\) and require that a particular word \(w\) appears only at the end of long “buffer” segments. By construction, the proportion \(N_w(n)/n\) then tends to \(0\) along certain scales rather than to \(b^{-k}\). Under any reasonable subnormality criterion that detects suppressed occurrences, such numbers qualify as subnormal.

4.1.2 Sparse expansions and growth-controlled patterns

Another construction uses a staging schedule: insert correct-looking blocks for a while, then switch to a deterministic low-complexity pattern for longer and longer intervals. If those intervals grow quickly enough, the cumulative discrepancy will exceed the threshold encoded by the subnormality definition. Conceptually, the digit sequence looks “nearly uniform” on short windows but fails in long-run averaging at a controlled rate.

4.2 Subnormal numbers defined via algorithmic rules

4.2.1 Cantor-like digit constraints

Cantor-type sets arise when digits are restricted to a sparse allowed set (e.g., only certain digits appear, or certain blocks are forbidden). The resulting numbers form a fractal subset of \([0,1]\). Since the empirical distribution of digits is forced to concentrate on a proper subset, digit-block frequencies cannot match uniform targets, producing subnormal behavior. Depending on the restriction strength, the number of different word frequencies tested can fail at different rates.

4.2.2 Iterative pattern substitution examples

Substitution systems replace symbols or blocks by longer blocks according to fixed rules. If the substitution creates long-range correlations—for example, by repeating certain marker patterns with rigid spacing—then occurrences of arbitrary test words become biased. When the substitution length grows in a way that amplifies these correlations, subnormality can be demonstrated by tracking how often target words appear in each substituted stage.

4.3 Comparing different bases through examples

Consider the same real number expressed in bases \(b\) and \(b'\). Even if the number is engineered to distort frequencies of certain base-\(b\) blocks, its base-\(b'\) expansion may partially “mix” digits differently due to base conversion. Illustrative examples often show:

  • subnormality in base \(b\) for a carefully chosen digit-constrained set,
  • but either weaker deviation or different failure modes in base \(b'\),

highlighting the representation dependence emphasized in the theory.

5 Analytical Tools and Methods

5.1 Techniques from uniform distribution theory

5.1.1 Weyl criterion and variants (contextual use)

Uniform distribution criteria translate questions about limiting frequencies into statements about averages of exponential functions. While the Weyl criterion is standard for equidistribution of sequences modulo 1, its “contextual use” here is to interpret digit statistics through exponential sums that respond to structured digit patterns. When digit blocks are constrained, these sums often fail to decay at the rates expected under normal-like behavior, yielding evidence for subnormality.

5.1.2 Discrepancy-based diagnostics

Discrepancy provides a quantitative measure of how far empirical distributions deviate from the uniform distribution on the space of digit blocks. One can compute or bound discrepancy from the construction:

  • if a word is rarely present, discrepancy grows because one cell in the partition carries too little mass;
  • if long deterministic segments dominate, discrepancy spikes around those segment endpoints.

These diagnostics directly align with subnormality’s requirement of non-maximal growth uniformity.

5.2 Measure-theoretic arguments

5.2.1 Borel–Cantelli type reasoning (overview)

When modeling digit sequences as random under the product measure, events describing “large deviations of block frequencies” can often be shown to have summable probabilities for certain thresholds. Borel–Cantelli reasoning then implies that almost surely only finitely many such deviation events occur, meaning true subnormality (as a strong deviation) is measure zero. Conversely, if probabilities are large enough or thresholds are adapted, one can identify regimes where infinitely many deviations occur—informing which definitions produce small or large subnormal sets.

In some frameworks, tail events—events determined by digits far out in the expansion—obey zero-one principles. Although the exact formulation depends on the definition of subnormality, this general idea helps explain why subnormal sets can have measure zero or full measure under certain probabilistic interpretations, and why the behavior of digit blocks is governed by asymptotic properties rather than finite prefixes.

5.3 Complexity/algorithmic perspectives

5.3.1 Growth rates and sublinear deviation measures

Subnormality criteria often encode growth comparisons: how quickly a deviation quantity must shrink (or how quickly it may grow). Analytical work focuses on proving lower bounds for deviation for constructed examples, or upper bounds for typical numbers. The central theme is the rate: normality corresponds to meeting the required decay, while subnormality corresponds to persistent failure in a sublinear deviation sense.

5.3.2 Pattern-recognition views of digit sequences

Another viewpoint treats digit expansions as strings whose statistical features can be detected by finite tests (e.g., counting specific blocks). Subnormal numbers are then those that fail certain uniformity tests—equivalently, those whose digit sequences are distinguishable from an ideal uniform model by empirical block-count statistics at long lengths. This connects naturally with “test-based” interpretations of randomness and bias.

6.1 Normal numbers and their contrasts

Normal numbers achieve uniform frequencies for all fixed block lengths. Subnormal numbers, by contrast, fail such uniformity in a manner strong enough to meet the chosen “non-maximal” growth/distribution definition. Thus, the distinction is not merely “not normal,” but “deviating in a structured rate-sensitive way.”

6.2 Champernowne-type constructions (high-level comparison)

Champernowne-type numbers are constructed by concatenating blocks from some systematic list (often integers in increasing order). These numbers can be normal in various settings, but the mechanism differs from digit-restriction approaches typical in subnormal constructions. From the standpoint of subnormality, Champernowne-like constructions illustrate that systematic concatenation may either promote uniformity or, if built with constrained orderings, can instead enforce bias.

6.3 Randomness and why subnormality indicates bias

If a number’s digits behaved like outputs of an unbiased random source, then long-run digit frequencies would concentrate near uniform targets. Subnormality functions as a certificate of non-randomness at the level of block statistics: it indicates an identifiable imbalance or a deviation that does not wash out quickly. This does not fully determine algorithmic randomness, but it reflects detectable bias under digit-frequency tests.

6.4 Fractal or dimension-based analogues

The sets defined by digit constraints often form fractal sets. In many cases, the “size” of the subnormal set is best expressed by dimension (e.g., Hausdorff dimension) rather than by measure alone. The relationship between subnormality and dimension arises because restrictions on digit patterns correspond to restricting branches in an iterated function system, producing non-trivial scaling laws.

7 Open Problems and Research Directions

7.1 Quantitative bounds on deviation from normality

A central direction is to sharpen how subnormality criteria translate into explicit quantitative bounds on discrepancy growth. Researchers aim to identify optimal thresholds: for a given base and deviation measure, what is the best possible rate separation between normal-like and subnormal-like behavior?

7.2 Base-invariance questions

Another line asks when subnormality in one base implies subnormality in another. While invariance generally fails without conditions, there are special cases—such as compatible base relationships or digit-grouping schemes—where partial invariance can be established. Determining the precise boundary of such transfer principles remains an active theme.

7.3 Effective (computable) characterizations

For algorithmic questions, one asks which subnormal numbers can be effectively described and how complex their digit constraints must be. Effective characterizations seek computable criteria that guarantee subnormality, and conversely, computable procedures that produce subnormal numbers with provable deviation rates.

Subnormality can be reframed through the lens of symbolic dynamics: digit sequences correspond to trajectories in a shift space defined by allowable words. Discrepancy minimization becomes the goal of analyzing which symbolic systems produce the most uniform (or least uniform) block distributions. This connects subnormality with broader investigations of how deterministic rule-based systems control statistical properties.

8 Practical Perspective (Non-controversial)

8.1 Empirical tests and heuristic experiments

In practice, one cannot verify subnormality directly from finitely many digits. Instead, experiments compute approximate block frequencies for lengths up to some limit and compare them to the uniform baseline. Heuristics then look for systematic drift beyond expected sampling fluctuations, which—while not conclusive—can strongly suggest non-normal behavior aligned with subnormality.

8.2 Interpreting finite-length evidence

Finite evidence must be interpreted carefully: early digit counts can fluctuate even for genuinely uniform-like sequences. Empirical signals of subnormality are stronger when they persist across multiple block lengths, multiple starting offsets, and repeated sampling windows, rather than appearing as a one-off anomaly.

8.3 Common pitfalls in digit-frequency estimation

Common errors include:

  • using ambiguous expansions (terminating versus non-terminating representations),
  • choosing block lengths that are too large for the available digit sample,
  • estimating frequencies with inconsistent sliding conventions,
  • ignoring carry/conversion artifacts when comparing across bases or derived representations.

Avoiding these pitfalls is essential for meaningful diagnostics.