Van der Corput Property in Mathematics

• The van der Corput property is a transdisciplinary concept defined by digit reversal methods that ensure low discrepancy and uniform distribution modulo 1.
• It unites approaches in quasi-Monte Carlo, ergodic theory, additive combinatorics, and spectral analysis by converting structured differences into global regularity.
• In analytic number theory, derivative tests based on the van der Corput method yield explicit bounds and facilitate cancellation in oscillatory sums.

Across the literature, the expression van der Corput property is used in several related senses attached to J. G. van der Corput’s sequence construction, difference theorem, and derivative method. In quasi-Monte Carlo and uniform distribution theory it denotes the combination of digit-reversal structure, uniform distribution modulo $1$, and low discrepancy typified by the base-$b$ radical-inverse sequence; in ergodic and semigroup settings it denotes a difference principle whereby control of correlations along shifts forces weak convergence or recurrence; in additive combinatorics it becomes a Fourier-analytic pseudorandomness property of sets of differences; and in analytic number theory it refers to derivative conditions that produce cancellation in exponential sums (Faure et al., 2015, Farhangi, 2021, Slijepcevic, 2010, Hiary, 2015).

1. Historical core and classical formulations

The classical van der Corput sequence is defined from the base-$b$ expansion

$$n=\sum_{k=0}^{M} a_k b^k,\qquad a_k\in\{0,\dots,b-1\},$$

by the radical-inverse map

$\phi_b(n)=\sum_{k=0}^{M} a_k b^{-(k+1)}.$

This digit reversal yields a sequence $Y_b=(\phi_b(n))_{n\ge 0}$ that is uniformly distributed modulo $1$, and in one dimension its star discrepancy has the optimal order $O((\log N)/N)$ up to constants; the 1935 binary construction is the prototype for later low-discrepancy digital sequences (Faure et al., 2015).

A second classical formulation is van der Corput’s Difference Theorem. In the form recalled by Farhangi, if $(x_n)_{n\ge1}\subset[0,1]$ is such that $(x_{n+h}-x_n)_{n\ge1}$ is uniformly distributed for every $b$0, then $b$1 is uniformly distributed (Farhangi, 2021). In this sense, the van der Corput property is not attached to a specific sequence, but to a transfer principle from differences to the original sequence.

These two classical strands already contain the enduring themes of the subject. One is constructive and digital: elementary base expansions generate sequences with strong equidistribution properties. The other is deductive and correlation-based: sufficiently random first differences force global equidistribution. Later work generalizes both themes.

2. Digital and discrepancy-theoretic meanings

In quasi-Monte Carlo theory, the van der Corput property is the conjunction of a digital construction rule and strong discrepancy bounds. The survey by Faure and Lemieux makes this explicit by treating the classical sequence, generalized van der Corput sequences $b$2 obtained by permuting digits, $b$3-sequences, Halton sequences, Hammersley point sets, and digital $b$4-sequences as successive extensions of the same $b$5-adic idea (Faure et al., 2015). In one dimension, the defining structural feature is that each $b$6-block of indices fills the $b$7 elementary intervals of length $b$8 in a controlled way; in higher dimensions, the analogue is exact occupancy of elementary boxes in $b$9-nets and $b$0-sequences.

Steiner extended the construction to abstract numeration systems $b$1, where $b$2 is an infinite regular language ordered by shortlex order. After defining a normalized value map $b$3 from admissible words to $b$4, the associated abstract van der Corput sequence is obtained by enumerating mirror words in the mirror language $b$5 and reading them back through $b$6. If $b$7 is recognized by a totally ordered Pisot automaton, then the resulting sequence satisfies

$b$8

so it is low discrepancy and hence uniformly distributed (0809.3994). Steiner also gave explicit discrepancy formulae and a characterization of bounded remainder sets $b$9 under slightly stronger automaton hypotheses (0809.3994).

A different one-dimensional generalization is given by Carbone’s LS-sequences. For integers $$n=\sum_{k=0}^{M} a_k b^k,\qquad a_k\in\{0,\dots,b-1\},$$0, $$n=\sum_{k=0}^{M} a_k b^k,\qquad a_k\in\{0,\dots,b-1\},$$1 with $$n=\sum_{k=0}^{M} a_k b^k,\qquad a_k\in\{0,\dots,b-1\},$$2, one fixes $$n=\sum_{k=0}^{M} a_k b^k,\qquad a_k\in\{0,\dots,b-1\},$$3 by $$n=\sum_{k=0}^{M} a_k b^k,\qquad a_k\in\{0,\dots,b-1\},$$4, refines intervals into $$n=\sum_{k=0}^{M} a_k b^k,\qquad a_k\in\{0,\dots,b-1\},$$5 long pieces of length $$n=\sum_{k=0}^{M} a_k b^k,\qquad a_k\in\{0,\dots,b-1\},$$6 and $$n=\sum_{k=0}^{M} a_k b^k,\qquad a_k\in\{0,\dots,b-1\},$$7 short pieces of length $$n=\sum_{k=0}^{M} a_k b^k,\qquad a_k\in\{0,\dots,b-1\},$$8, and then reorders left endpoints. The digit algorithm uses base $$n=\sum_{k=0}^{M} a_k b^k,\qquad a_k\in\{0,\dots,b-1\},$$9, an admissibility condition excluding forbidden digit transitions, and an LS-radical inverse

$\phi_b(n)=\sum_{k=0}^{M} a_k b^{-(k+1)}.$0

Theorem 5.3 identifies the LS-sequence of points with $\phi_b(n)=\sum_{k=0}^{M} a_k b^{-(k+1)}.$1, and when $\phi_b(n)=\sum_{k=0}^{M} a_k b^{-(k+1)}.$2 the discrepancy satisfies

$\phi_b(n)=\sum_{k=0}^{M} a_k b^{-(k+1)}.$3

which is the same order as the classical van der Corput sequence (Carbone, 2013). The special case $\phi_b(n)=\sum_{k=0}^{M} a_k b^{-(k+1)}.$4, $\phi_b(n)=\sum_{k=0}^{M} a_k b^{-(k+1)}.$5 recovers the base-$\phi_b(n)=\sum_{k=0}^{M} a_k b^{-(k+1)}.$6 van der Corput sequence exactly (Carbone, 2013).

Recent work also shows that the digital van der Corput component can participate in genuinely low-discrepancy hybrids. Robertson proved that for an irreducible polynomial $\phi_b(n)=\sum_{k=0}^{M} a_k b^{-(k+1)}.$7 and a Laurent series $\phi_b(n)=\sum_{k=0}^{M} a_k b^{-(k+1)}.$8 induced from a counterexample to the $\phi_b(n)=\sum_{k=0}^{M} a_k b^{-(k+1)}.$9-adic Littlewood conjecture, the two-dimensional hybrid

$Y_b=(\phi_b(n))_{n\ge 0}$0

satisfies

$Y_b=(\phi_b(n))_{n\ge 0}$1

providing an explicit low-discrepancy digital Kronecker–van der Corput hybrid (Robertson, 2024).

3. Fine-scale distribution and probabilistic refinements

The discrepancy-theoretic van der Corput property does not imply random local spacing. Wohlfarter derived an explicit formula for the finite empirical pair correlation function $Y_b=(\phi_b(n))_{n\ge 0}$2 of the base-$Y_b=(\phi_b(n))_{n\ge 0}$3 van der Corput sequence and showed that $Y_b=(\phi_b(n))_{n\ge 0}$4 exists only for $Y_b=(\phi_b(n))_{n\ge 0}$5, where the limit equals $Y_b=(\phi_b(n))_{n\ge 0}$6 (Weiß, 2023). This gives a sharply non-Poissonian local picture: on the $Y_b=(\phi_b(n))_{n\ge 0}$7 scale, the base-$Y_b=(\phi_b(n))_{n\ge 0}$8 sequence exhibits strong repulsion rather than the Poisson law $Y_b=(\phi_b(n))_{n\ge 0}$9.

At the same time, additive functionals of the sequence admit probabilistic limit laws. For the base-$1$0 van der Corput sequence $1$1, Drmota, Larcher, and Pillichshammer study

$1$2

and prove a central limit theorem with explicit error term, together with a large deviation estimate. The normalizing constants are

$1$3

and the same asymptotic law extends to the $1$4-discrepancy for $1$5 (Borda, 2016). Thus the classical one-dimensional van der Corput sequence is simultaneously rigid at the pair-correlation scale and amenable to Gaussian fluctuation theory for integrated discrepancy functionals.

This contrast is structurally important. It indicates that the van der Corput property in the low-discrepancy sense governs coarse and mesoscopic equidistribution, but does not enforce Poissonian fine-scale statistics. A plausible implication is that discrepancy-optimal digital sequences should be analyzed separately from random-like local-spacing models.

4. Additive-combinatorial and Fourier-analytic formulations

In additive combinatorics, a set $1$6 is called a van der Corput set if, for every real sequence $1$7, uniform distribution of every difference sequence $1$8 with $1$9 implies uniform distribution of $O((\log N)/N)$0 itself (Slijepcevic, 2010). Kamae–Mendès France and Ruzsa showed that this is equivalent to a Fourier-analytic positivity criterion: if $O((\log N)/N)$1 denotes the normed nonnegative cosine polynomials with spectrum in $O((\log N)/N)$2, then

$O((\log N)/N)$3

tends to $O((\log N)/N)$4 precisely when $O((\log N)/N)$5 is a van der Corput set (Slijepcevic, 2010).

For the set of perfect squares $O((\log N)/N)$6, Slijepčević proved the first published quantitative upper bound

$O((\log N)/N)$7

by constructing nonnegative normed cosine polynomials with spectrum in the squares up to $O((\log N)/N)$8 and small constant term $O((\log N)/N)$9 (Slijepcevic, 2010). This answers a problem of Ruzsa and Montgomery in that case and shows that the van der Corput property of squares can be made quantitative.

A stronger modern formulation views the van der Corput property as a pseudorandomness statement about a finite difference set. For

$(x_n)_{n\ge1}\subset[0,1]$0

Green and Walker proved that for every $(x_n)_{n\ge1}\subset[0,1]$1 there exist coefficients $(x_n)_{n\ge1}\subset[0,1]$2, supported on $(x_n)_{n\ge1}\subset[0,1]$3, with $(x_n)_{n\ge1}\subset[0,1]$4 and

$(x_n)_{n\ge1}\subset[0,1]$5

As a consequence, any $(x_n)_{n\ge1}\subset[0,1]$6 with $(x_n)_{n\ge1}\subset[0,1]$7 satisfies

$(x_n)_{n\ge1}\subset[0,1]$8

a power-saving Sárközy-type theorem for sums of two squares (Fan et al., 28 Jun 2026).

An analogous function-field version was obtained for shifted irreducibles. In $(x_n)_{n\ge1}\subset[0,1]$9, if $(x_{n+h}-x_n)_{n\ge1}$0 contains no pair with difference $(x_{n+h}-x_n)_{n\ge1}$1 for irreducible $(x_{n+h}-x_n)_{n\ge1}$2, then

$(x_{n+h}-x_n)_{n\ge1}$3

The proof relies on a nonnegative cosine polynomial supported on $(x_{n+h}-x_n)_{n\ge1}$4 whose constant term is $(x_{n+h}-x_n)_{n\ge1}$5, described explicitly as the van der Corput property for shifted irreducibles (Fan et al., 31 Oct 2025).

5. Ergodic, semigroup, and spectral generalizations

A broad abstract version of the van der Corput property was developed by Tserunyan for actions of semigroups along filters. Given a filter $(x_{n+h}-x_n)_{n\ge1}$6 on a semigroup $(x_{n+h}-x_n)_{n\ge1}$7, the paper defines differentiation of subsets by

$(x_{n+h}-x_n)_{n\ge1}$8

introduces $(x_{n+h}-x_n)_{n\ge1}$9-filters as those respecting the resulting higher-order differentiation calculus, and proves a difference-Ramsey theorem for graphs on $b$00 with edges labelled by ratios (Tserunyan, 2014). The central consequence is a general van der Corput lemma: for a bounded weakly upper semimeasurable Hilbert-space-valued sequence $b$01,

$b$02

This subsumes previously known cases for density filters, IP$b$03-filters, idempotent ultrafilters, and conull filters of invariant measures (Tserunyan, 2014).

Farhangi recast the difference theorem in spectral language. For sequences $b$04, the paper defines sL-sequences by requiring that every mean-zero continuous observable $b$05 have Lebesgue spectral measure, wm-sequences by requiring weak-mixing-type behavior, and o-sequences by requiring near orthogonality (Farhangi, 2021). Two general discrepancy criteria are proved: if

$b$06

then $b$07 is an sL-sequence; and if

$b$08

then $b$09 is a wm-sequence (Farhangi, 2021).

The same framework yields new subsequence and pair-distribution results. If $b$10 is such that all difference sequences are uniformly distributed, then the subsequence indexed by the positions of the $b$11s in the classical Thue–Morse sequence is uniformly distributed as well (Farhangi, 2021). Farhangi also proved that $b$12 is an o-sequence if and only if, for every $b$13, the pair sequence $b$14 is uniformly distributed in $b$15 (Farhangi, 2021). In this spectral setting, the van der Corput property becomes a hierarchy of difference criteria corresponding to Lebesgue, continuous, discrete, or singular spectral types.

6. Analytic-number-theoretic and oscillatory-integral formulations

In analytic number theory, van der Corput’s method is a derivative test for exponential sums. Hiary gave an explicit third-derivative version: if $b$16 and

$b$17

then

$b$18

with explicit $b$19 (Hiary, 2015). Applied to the Riemann–Siegel sum for $b$20, this yields

$b$21

an explicit van der Corput-type bound on the critical line (Hiary, 2015).

Arias de Reyna generalized this to explicit $b$22-th derivative estimates. If $b$23 has $b$24 continuous derivatives on $b$25 and

$b$26

then, with $b$27,

$b$28

where $b$29 are explicit and satisfy $b$30, $b$31, $b$32 for $b$33 (Reyna, 2024). The paper also corrects an error in van der Corput’s 1937 induction argument and proves that the explicit theorem implies Titchmarsh’s classical Theorem 5.13 (Reyna, 2024).

A higher-dimensional analogue has recently been developed for oscillatory integrals. Using toric resolution adapted to the Newton polyhedron of a real-analytic phase $b$34, one reduces the phase locally to monomials and then applies a one-dimensional van der Corput argument in suitable coordinates. For

$b$35

with $b$36 $b$37-nondegenerate, the decay estimate is

$b$38

if $b$39 is not an integer, and

$b$40

if $b$41 is an integer, where $b$42 is the Newton distance and $b$43 is the codimension of the principal face (Xu, 25 Nov 2025). Here the van der Corput property is governed by Newton-polyhedral geometry rather than by a single directional derivative.

The name also appears in inequality theory. Baricz, Jankov Maširević, and Pogány showed that if $b$44 is positive and log-concave, then

$b$45

and used this mechanism to extend van der Corput inequalities from $b$46 and $b$47 to $b$48, $b$49, and $b$50 Bessel families on intervals where the relevant derivatives are log-concave (Baricz et al., 2014). This is a different but historically connected usage: the van der Corput property is the endpoint-geometric-mean lower bound induced by derivative log-concavity.

Taken together, these formulations show that the van der Corput property is best understood as a transdisciplinary principle: digit reversal yields optimal equidistribution; small or structured differences force global uniformity or recurrence; Fourier nonnegativity of difference sets yields Sárközy-type theorems; and derivative nondegeneracy yields cancellation in oscillatory sums and integrals. The common thread is the conversion of local structure—digital, additive, spectral, or differential—into global regularity.