Van der Corput Sequence

• The van der Corput sequence is defined by reflecting the digits of nonnegative integers in a fixed base to produce a uniformly distributed sequence in [0, 1).
• It achieves a low discrepancy of order O((log N)/N) and is fundamental in quasi-Monte Carlo integration and uniform distribution theory.
• Generalizations such as LS-sequences and σ-scrambled variants extend its methodology to advanced numerical methods and higher-dimensional sampling.

The van der Corput sequence is a prototypical one-dimensional low-discrepancy sequence constructed by reflecting the digits of nonnegative integers in a fixed base and interpreting the result as a $b$-adic fraction. This sequence forms the foundation of much of modern quasi-Monte Carlo theory and uniform distribution practice, combining elegant digital construction, rigorous discrepancy bounds, deep links to harmonic analysis, and wide generalization to other algebraic and combinatorial settings. The following provides a comprehensive account of its construction, properties, generalizations, and significance in mathematical research.

1. Definition and Classical Construction

Let $b \geq 2$ be an integer base. Every nonnegative integer $n$ admits a unique base-$b$ expansion: $$n = \sum_{i=0}^{m} a_i b^i, \quad a_i \in \{0, 1, \dots, b-1\}.$$ The radical-inverse function $\phi_b : \mathbb{N} \to [0,1)$ is defined by reversing the order of digits: $\phi_b(n) = \sum_{i=0}^m a_i b^{-(i+1)}.$ In positional notation, if $[n]_b = a_m a_{m-1} \dots a_0$, then $\phi_b(n) = 0.a_0 a_1 \dots a_m$ in base $b$.

The van der Corput sequence in base $b$ is given by $(\phi_b(n))_{n=0}^\infty$ (Carbone, 2013, Faure et al., 2015). For example, in base $2$:

• $n = 3$ ($[3]_2 = 11$): $\phi_2(3) = 0.11_2 = 3/4$.

The process can be efficiently implemented using the pseudocode:

def van_der_corput(n, b): x = 0 f = 1 / b while n > 0: a = n % b x += a * f n //= b f /= b return x

(Carbone, 2013)

2. Geometric Interpretation and Digit Inversion

The digit inversion central to van der Corput's construction has a geometric explanation. Define the contraction-and-shift maps: $$V_i(x) = \frac{1}{b} x + \frac{i}{b}, \quad i \in \{0, \dots, b-1\}.$$ Then

$$\phi_b(n) = V_{a_0} \circ V_{a_1} \circ \cdots \circ V_{a_m}(0).$$

Starting from $[0,1)$, repeatedly zoom into subintervals based on each digit $a_i$, applying the maps in least-significant digit order; this "magical" digit order inversion is both algorithmically essential and geometrically insightful (Carbone, 2013).

3. Discrepancy, Uniform Distribution, and Lebesgue Constants

A sequence $(x_n)$ in $[0,1)$ is uniformly distributed mod 1 if, for all $0 \leq a < b \leq 1$,

$$\lim_{N \to \infty} \frac{1}{N} \#\{0 \leq n < N : x_n \in [a,b)\} = b - a.$$

The star discrepancy $D_N^*$ quantifies the maximal deviation from true uniformity: $$D_N^* = \sup_{u \in [0,1]} \left| \frac{1}{N} \#\{ n < N : x_n < u \} - u \right|.$$ For the classical van der Corput sequence,

$D_N^*(\phi_b) = O\left(\frac{\log N}{N}\right),$

this bound being optimal for $b$-adic constructions (Faure et al., 2015, Hofer, 20 Jan 2025).

In base $2$, the normalized star discrepancy coincides with the Lebesgue constants of the Walsh system: $N D_N^*(\{\phi_2(n)\}_{n=0}^{N-1}) = \Lambda_N,$ where $\Lambda_N$ is the Lebesgue constant of the $N$th Walsh-Dirichlet kernel. This coincidence creates a direct bridge between uniform distribution theory and approximation theory, allowing transfer of exact and asymptotic results (Dick et al., 4 Dec 2024).

4. $L_p$-Discrepancy and Davenport Symmetrization

For $p \in [1, \infty)$, the $L_p$-discrepancy of the van der Corput sequence is: $$\left\| D_N(\phi_b, \cdot) \right\|_{L_p} = \left( \int_0^1 |D_N(\phi_b, t)|^p dt \right)^{1/p}.$$ Standard van der Corput sequences satisfy: $$\left\| D_N(\phi_b, \cdot) \right\|_{L_p} = O\left(\frac{\log N}{N}\right),$$ but Roth and Proinov established that no sequence can do better than $c_p \sqrt{\log N}/N$ infinitely often (Kritzinger et al., 2015, Kritzinger, 2015).

Davenport's symmetrization (reflection principle) produces: $$z_{2m} = \phi_b(m), \qquad z_{2m+1} = 1 - \phi_b(m), \quad m \ge 0,$$ yielding

$$\left\| D_N(\{z_n\}, \cdot) \right\|_{L_p} = O\left(\frac{\sqrt{\log N}}{N}\right)$$

for all $1 < p < \infty$, which is best possible (Kritzinger et al., 2015, Kritzinger, 2015).

5. Pair Correlations and Rigidity

The finite empirical pair correlation function,

$$F_N(s) = \frac{1}{N} \# \left\{ 1 \le i \neq j \le N : \|x_i - x_j\| \le \frac{s}{N} \right\},$$

was computed explicitly in base $2$ in terms of binary coefficients of $N$ (Weiß, 2023). As $N \to \infty$,

$$\lim_{N \to \infty} F_N(s) = 0, \quad \text{for } 0 \leq s \leq 1/2,$$

indicating non-Poissonian pair correlations: points do not cluster at the $1/N$ scale and the gap structure is extremely rigid.

6. Generalizations: LS-Sequences, Abstract Numeration, Digit Permutation

Carbone (Carbone, 2013, Carbone, 2012) and subsequent works define LS-sequences as a two-parameter $(L,S)$ generalization: natural numbers are expanded in base $B = L + S$, digits are inverted, and each digit index is weighted by self-similar contraction maps. The classical van der Corput sequence is recovered for $L = b$, $S = 0$.

Further, the concept generalizes to abstract numeration systems recognized by totally ordered Pisot automata (0809.3994). Abstract van der Corput sequences constructed in such regular languages retain uniform distribution and $O(\log N)$ discrepancy, provided automata satisfy spectral gap (Pisot) properties.

Permuted van der Corput sequences ($\sigma$-scramblings) and higher-dimensional digital analogues (Halton, Niederreiter $(t,s)$-sequence, digital nets) inherit and generalize these principles, serving as core constructions for quasi-Monte Carlo integration (Faure et al., 2015, Hofer, 20 Jan 2025).

7. Connections: Potential Theory, Randomness, and Further Directions

Pausinger (Pausinger, 2019) established that the van der Corput sequence arises as the output of a greedy energy-minimization system under strictly convex symmetric kernels, including the logarithmic interaction. This links deterministic low-discrepancy sequence generation to optimal point distributions in potential theory and Leja sequences.

Randomness properties are also quantifiable: sums over the van der Corput sequence exhibit central limit theorems and large deviation inequalities, with fluctuations about the mean governed by explicit expressions in terms of digit structure and Fourier coefficients (Borda, 2016, Dick et al., 4 Dec 2024).

The van der Corput sequence remains central, not only in uniform distribution/algebraic combinatorics but also in discrete harmonic analysis, probabilistic limit theory, and energy optimization. Open problems involve multidimensional pair correlations, optimal constant determination, distribution properties for nonstandard numeration systems, and dimension-robust generalization in high-dimensional net construction (Weiß, 2023, 0809.3994, Dick et al., 4 Dec 2024, Hofer, 20 Jan 2025).

• "How to construct generalized van der Corput sequences" (Carbone, 2013)
• "On the finite pair correlation function of van der Corput sequences" (Weiß, 2023)
• "$L_p$-discrepancy of the symmetrized van der Corput sequence" (Kritzinger et al., 2015)
• "On the exact order of the discrepancy of low discrepancy digital van der Corput--Kronecker sequences" (Hofer, 20 Jan 2025)
• "Greedy energy minimization can count in binary: point charges and the van der Corput sequence" (Pausinger, 2019)
• "Discrepancy results for the Van der Corput sequence" (Spiegelhofer, 2017)
• "Regularities of the distribution of abstract van der Corput sequences" (0809.3994)
• "A van der Corput-type algorithm for LS-sequences of points" (Carbone, 2012)
• "From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules" (Faure et al., 2015)
• "Lebesgue constants for the Walsh system and the discrepancy of the van der Corput sequence" (Dick et al., 4 Dec 2024)
• "L_p- and S_{p,q}^rB-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases" (Kritzinger, 2015)
• "On the distribution of the van der Corput sequence in arbitrary base" (Borda, 2016)