---
title: Van der Corput Property in Mathematics
url: https://www.emergentmind.com/topics/van-der-corput-property
type: topic
---

# Van der Corput Property in Mathematics

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 [1506.03764] [2106.01123] [1003.3780] [1507.01261].

## 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 [1506.03764].

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 \(h\in\mathbb N\), then \((x_n)_{n\ge1}\) is uniformly distributed [2106.01123]. 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 \(Y_b^\Sigma\) obtained by permuting digits, \((0,1)\)-sequences, Halton sequences, Hammersley point sets, and digital \((t,s)\)-sequences as successive extensions of the same \(b\)-adic idea [1506.03764]. In one dimension, the defining structural feature is that each \(b^m\)-block of indices fills the \(b^m\) elementary intervals of length \(b^{-m}\) in a controlled way; in higher dimensions, the analogue is exact occupancy of elementary boxes in \((t,m,s)\)-nets and \((t,s)\)-sequences.

Steiner extended the construction to **abstract numeration systems** \(S=(L,A,\le)\), where \(L\subseteq A^*\) is an infinite regular language ordered by shortlex order. After defining a normalized value map \(\langle\cdot\rangle\) from admissible words to \([0,1]\), the associated abstract van der Corput sequence is obtained by enumerating mirror words in the mirror language \(\widetilde{L'}\) and reading them back through \(\langle\cdot\rangle\). If \(L\) is recognized by a totally ordered Pisot automaton, then the resulting sequence satisfies
\[
\sup_{I\subseteq[0,1)} |D(N,I)|=\mathcal O(\log N),
\]
so it is low discrepancy and hence uniformly distributed [0809.3994]. Steiner also gave explicit discrepancy formulae and a characterization of bounded remainder sets \([0,y)\) under slightly stronger automaton hypotheses [0809.3994].

A different one-dimensional generalization is given by Carbone’s LS-sequences. For integers \(L\ge1\), \(S\ge0\) with \(L+S>2\), one fixes \(\gamma\in(0,1)\) by \(L\gamma+S\gamma^2=1\), refines intervals into \(L\) long pieces of length \(\gamma\) and \(S\) short pieces of length \(\gamma^2\), and then reorders left endpoints. The digit algorithm uses base \(L+S\), an admissibility condition excluding forbidden digit transitions, and an LS-radical inverse
\[
\phi_{L,S}(n)=\sum_{k=0}^{M}\tilde a_k(n)\gamma^{k+1}.
\]
Theorem 5.3 identifies the LS-sequence of points with \(\{\phi_{L,S}(n)\}_{n\in N_{L,S}}\), and when \(S\le L\) the discrepancy satisfies
\[
N D(Q_N)\le k_1\log N,
\]
which is the same order as the classical van der Corput sequence [1304.5083]. The special case \(L=b\), \(S=0\) recovers the base-\(b\) van der Corput sequence exactly [1304.5083].

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 \(P(t)\in\mathbb F_q[t]\) and a Laurent series \(\Phi(t)\) induced from a counterexample to the \(t\)-adic Littlewood conjecture, the two-dimensional hybrid
\[
\mathbf H(\Phi,P)=\bigl(K_n(\Phi),V_n(P)\bigr)_{n\ge0}
\]
satisfies
\[
D_N^*(\mathbf H(\Phi,P))\ll_{\Theta,P}\frac{\log^2 N}{N},
\]
providing an explicit low-discrepancy digital Kronecker–van der Corput hybrid [2409.05469].

## 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 \(F_N(s)\) of the base-\(2\) van der Corput sequence and showed that \(\lim_{N\to\infty}F_N(s)\) exists only for \(0\le s\le 1/2\), where the limit equals \(0\) [2306.10437]. This gives a sharply non-Poissonian local picture: on the \(1/N\) scale, the base-\(2\) sequence exhibits strong repulsion rather than the Poisson law \(2s\).

At the same time, additive functionals of the sequence admit probabilistic limit laws. For the base-\(b\) van der Corput sequence \(I_n\), Drmota, Larcher, and Pillichshammer study
\[
S(N)=\sum_{n=0}^{N-1}\Bigl(\tfrac12-I_n\Bigr)=\int_0^1 A_N(x)\,dx
\]
and prove a central limit theorem with explicit error term, together with a large deviation estimate. The normalizing constants are
\[
c(b)=\frac{b^2-1}{12b},\qquad d(b)=\frac{b^4+120b^3-480b^2+600b-241}{720b^2},
\]
and the same asymptotic law extends to the \(L^p\)-discrepancy for \(1<p<\infty\) [1606.07944]. 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 \(D\subset\mathbb N\) is called a **van der Corput set** if, for every real sequence \((c_n)\), uniform distribution of every difference sequence \((c_{n+d}-c_n)\) with \(d\in D\) implies uniform distribution of \((c_n)\) itself [1003.3780]. Kamae–Mendès France and Ruzsa showed that this is equivalent to a Fourier-analytic positivity criterion: if \(\mathcal T(D_n)\) denotes the normed nonnegative cosine polynomials with spectrum in \(D\cap\{1,\dots,n\}\), then
\[
\gamma(n):=\inf_{T\in\mathcal T(D_n)} a_0
\]
tends to \(0\) precisely when \(D\) is a van der Corput set [1003.3780].

For the set of perfect squares \(Q=\{m^2:m\in\mathbb N\}\), Slijepčević proved the first published quantitative upper bound
\[
\gamma(n)=O\bigl((\log n)^{-1/3}\bigr)
\]
by constructing nonnegative normed cosine polynomials with spectrum in the squares up to \(n\) and small constant term \(a_0\) [1003.3780]. 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
\[
S_N=\{1\le d\le N:d=x^2+y^2\text{ for some }x,y\in\mathbb Z\},
\]
Green and Walker proved that for every \(\varepsilon>0\) there exist coefficients \(c_d\ge0\), supported on \(S_N\), with \(\sum_{d\in S_N}c_d=1\) and
\[
\Re\sum_{d\in S_N} c_d e(d\theta)\ge -C_\varepsilon N^{-1/8+\varepsilon}
\qquad(\theta\in\mathbb R/\mathbb Z).
\]
As a consequence, any \(A\subseteq[N]\) with \((A-A)\cap S_N=\varnothing\) satisfies
\[
|A|\ll_\varepsilon N^{7/8+\varepsilon},
\]
a power-saving Sárközy-type theorem for sums of two squares [2606.29185].

An analogous function-field version was obtained for shifted irreducibles. In \(\mathbb F_q[t]\), if \(A\subseteq\{f:\deg f\le N\}\) contains no pair with difference \(P-1\) for irreducible \(P\), then
\[
|A|\ll_{q,\varepsilon} q^{(N+1)(11/12+\varepsilon)}.
\]
The proof relies on a nonnegative cosine polynomial supported on \(\{0\}\cup\{P-1\}\) whose constant term is \(\ll \widehat N^{-1/12+\varepsilon}\), described explicitly as the van der Corput property for shifted irreducibles [2510.27581].

## 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 \(\mathcal F\) on a semigroup \(G\), the paper defines differentiation of subsets by
\[
\partial_g A:=A\cap A g^{-1},
\]
introduces \(\partial\)-filters as those respecting the resulting higher-order differentiation calculus, and proves a difference-Ramsey theorem for graphs on \(G\) with edges labelled by ratios [1411.3262]. The central consequence is a general van der Corput lemma: for a bounded weakly upper semimeasurable Hilbert-space-valued sequence \((e_g)\),
\[
\lim_{h\to\mathcal F}\lim_{g\to\mathcal F}\langle e_g,e_{gh}\rangle=0
\quad\Longrightarrow\quad
\lim_{g\to\mathcal F}\langle f,e_g\rangle=0\ \text{ for all }f.
\]
This subsumes previously known cases for density filters, IP\(^*\)-filters, idempotent ultrafilters, and conull filters of invariant measures [1411.3262].

Farhangi recast the difference theorem in spectral language. For sequences \((x_n)\subset[0,1]^d\), the paper defines **sL-sequences** by requiring that every mean-zero continuous observable \(f(x_n)\) have Lebesgue spectral measure, **wm-sequences** by requiring weak-mixing-type behavior, and **o-sequences** by requiring near orthogonality [2106.01123]. Two general discrepancy criteria are proved: if
\[
\sum_{h=1}^{\infty}\overline D\bigl((x_{n+h}-x_n)_{n\ge1}\bigr)^2<\infty,
\]
then \((x_n)\) is an sL-sequence; and if
\[
\lim_{H\to\infty}\frac1H\sum_{h=1}^H \overline D\bigl((x_{n+h}-x_n)_{n\ge1}\bigr)=0,
\]
then \((x_n)\) is a wm-sequence [2106.01123].

The same framework yields new subsequence and pair-distribution results. If \((x_n)\) is such that all difference sequences are uniformly distributed, then the subsequence indexed by the positions of the \(1\)s in the classical Thue–Morse sequence is uniformly distributed as well [2106.01123]. Farhangi also proved that \((x_n)\) is an o-sequence if and only if, for every \(h\in\mathbb N\), the pair sequence \((x_n,x_{n+h})\) is uniformly distributed in \([0,1]^{2d}\) [2106.01123]. 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 \(f\in C^3[N+1,N+L]\) and
\[
\frac1W\le |f'''(x)|\le \frac{\lambda}{W},
\]
then
\[
\Bigl|\sum_{n=N+1}^{N+L} e^{2\pi i f(n)}\Bigr|^2
\le
(LW^{-1/3}+\eta)\bigl(\alpha L+\beta W^{2/3}\bigr),
\]
with explicit \(\alpha,\beta\) [1507.01261]. Applied to the Riemann–Siegel sum for \(\zeta(1/2+it)\), this yields
\[
|\zeta(1/2+it)|\le 0.63\, t^{1/6}\log t \qquad (t\ge3),
\]
an explicit van der Corput-type bound on the critical line [1507.01261].

Arias de Reyna generalized this to explicit \(d\)-th derivative estimates. If \(f\) has \(d\) continuous derivatives on \((X,X+Y]\) and
\[
0<\lambda\le f^{(d)}(x)\le\Lambda,
\]
then, with \(D=2^d\),
\[
\Bigl|\frac1Y\sum_{X<n\le X+Y} e(f(n))\Bigr|
\le
\max\Biggl\{
A_d\Bigl(\frac{\Lambda}{\lambda Y}\Bigr)^{2/D},
B_d\Bigl(\frac{\Lambda^2}{\lambda}\Bigr)^{1/(D-2)},
C_d(\lambda Y^d)^{-2/D}
\Biggr\},
\]
where \(A_d,B_d,C_d\) are explicit and satisfy \(A_d<7.5\), \(B_d<5.8\), \(C_d<10.9\) for \(d\ge2\) [2407.02094]. 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 [2407.02094].

A higher-dimensional analogue has recently been developed for oscillatory integrals. Using toric resolution adapted to the Newton polyhedron of a real-analytic phase \(f\), one reduces the phase locally to monomials and then applies a one-dimensional van der Corput argument in suitable coordinates. For
\[
I(\lambda)=\int_{\mathbb R^n} e^{i\lambda f(x)}\varphi(x)\,dx,
\]
with \(f\) \(\mathbb R\)-nondegenerate, the decay estimate is
\[
|I(\lambda)|\le C_\varphi \lambda^{-1/d_f}(\log\lambda)^{k-1}
\]
if \(1/d_f\) is not an integer, and
\[
|I(\lambda)|\le C_\varphi \lambda^{-1/d_f}(\log\lambda)^k
\]
if \(1/d_f\) is an integer, where \(d_f\) is the Newton distance and \(k\) is the codimension of the principal face [2511.19922]. 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 \(f'\) is positive and log-concave, then
\[
|f(b)-f(a)|\ge |b-a|\sqrt{f'(a)f'(b)},
\]
and used this mechanism to extend van der Corput inequalities from \(\cos\) and \(\cosh\) to \(J_\nu\), \(I_\nu\), and \(K_\nu\) Bessel families on intervals where the relevant derivatives are log-concave [1401.7764]. 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.

Source: https://www.emergentmind.com/topics/van-der-corput-property