---
title: 'Cusick Conjecture: Binary Sum-of-Digits'
url: https://www.emergentmind.com/topics/cusick-conjecture
type: topic
---

# Cusick Conjecture: Binary Sum-of-Digits

The Cusick conjecture is a statement about the binary sum-of-digits function under translation. For a nonnegative integer \(t\), let \(s(n)\) denote the number of \(1\)'s in the base-\(2\) expansion of \(n\), and define
\[
c_t=\lim_{N\to\infty}\frac1N\bigl|\{n<N:s(n+t)\ge s(n)\}\bigr|,
\]
where the limit exists. T. W. Cusick conjectured that \(c_t>\frac12\) for every \(t\ge0\). The conjecture connects carry propagation in binary addition, valuations of rising factorials and binomial coefficients, and asymptotic densities for digital functions. A partial resolution showed that \(c_t>\frac12\) for \(t\) in a set of asymptotic density \(1\), and a later first-exit argument proved the conjecture in full with the explicit bound
\[
c_t\ge \frac12+2^{-2s_2(t)-1}
\]
for every \(t\ge1\) [1509.08623] [2606.23398].

## 1. Definitions and basic distributions

Let
\[
n=\sum_{i\ge0}\varepsilon_i2^i,\qquad \varepsilon_i\in\{0,1\},
\]
and define
\[
s(n)=\sum_{i\ge0}\varepsilon_i.
\]
For each fixed \(t\ge0\), the central density is
\[
c_t=\lim_{N\to\infty}\frac1N\bigl|\{n<N:s(n+t)\ge s(n)\}\bigr|.
\]
A closely related quantity is
\[
\tilde c_t=\lim_{N\to\infty}\frac1N\bigl|\{n<N:s(n+t)>s(n)\}\bigr|.
\]

A finer distribution is obtained by recording the exact digit-sum difference. One introduces
\[
\delta(k,t)=\operatorname{dens}\{n:s(n+t)-s(n)=k\},
\]
or equivalently, in later notation,
\[
\mu_t(d)=\lim_{N\to\infty}\frac1N\bigl|\{n<N:s(n+t)-s(n)=d\}\bigr|.
\]
These densities define probability measures on \(\mathbb Z\); in the \(\mu_t\) formulation they have mean zero. The tails recover the main quantities:
\[
c_t=\sum_{k\ge0}\delta(k,t),\qquad \tilde c_t=\sum_{k>0}\delta(k,t).
\]

The distributions satisfy binary dilation and odd-step recurrences. In the \(\delta\)-notation,
\[
\delta(k,2t)=\delta(k,t),\qquad
\delta(k,2t+1)=\tfrac12\delta(k-1,t)+\tfrac12\delta(k+1,t+1).
\]
In the \(\mu_t\)-notation,
\[
\mu_{2t}(d)=\mu_t(d),\qquad
\mu_{2t+1}(d)=\tfrac12\,\mu_t(d-1)+\tfrac12\,\mu_{t+1}(d+1).
\]
These recurrences are the basic structural relations underlying most later work [1509.08623] [2605.08624].

## 2. Equivalent formulations

A striking feature of the conjecture is that it admits several equivalent formulations.

The first is in terms of carry-counting in binary addition. If \(\nu_2(m)\) denotes the exponent of \(2\) dividing \(m\), and
\[
(x)_t=x(x+1)\cdots(x+t-1)
\]
is the rising factorial, then
\[
s(n+t)-s(n)=t-\nu_2((n+1)_t).
\]
Hence
\[
s(n+t)\ge s(n)
\quad\Longleftrightarrow\quad
2^{t+1}\nmid (n+1)_t.
\]
Because \(n\mapsto (n+1)_t \bmod 2^{t+1}\) is periodic of period \(2^{t+1}\), the density \(c_t\) can be interpreted through the zero set of a polynomial modulo a power of \(2\).

A second formulation uses Pascal’s triangle. From Legendre’s formula \(s(t)!=t-s(t)\), one obtains
\[
s(n+t)-s(n)=s(t)-\nu_2\!\binom{n+t}{t}.
\]
Thus
\[
c_t=\operatorname{dens}\{n:2^{s(t)+1}\nmid \binom{n+t}{t}\}.
\]
Equivalently,
\[
X(X+1)\cdots(X+t-1)
\]
has fewer than \(2^t\) zeros modulo \(2^{t+1}\).

These reformulations place the conjecture simultaneously in the theory of digital sums, \(2\)-adic valuations, and arithmetic properties of binomial coefficients and polynomial congruences [1509.08623].

## 3. The density-one theorem and analytic-combinatorial proof

The first major progress was obtained by Drmota, Kauers, and Spiegelhofer. Their main theorem states that for any \(\varepsilon>0\), as \(T\to\infty\),
\[
\bigl|\{t<T:\tfrac12-\varepsilon<\tilde c_t<\tfrac12<c_t<\tfrac12+\varepsilon\}\bigr|
= T+O(T/\ln T).
\]
In particular, the set of \(t\) for which \(c_t>\frac12\) has asymptotic density \(1\). Simultaneously, \(\tilde c_t<\frac12\) on a set of density \(1\).

The proof begins with dyadic blocks \(t\in[2^\lambda,2^{\lambda+1})\). Writing
\[
m_{k,\lambda}=\frac1{2^\lambda}\sum_{2^\lambda\le t<2^{\lambda+1}}\delta(k,t),
\]
one obtains an explicit formula for \(m_{k,\lambda}\) by unfolding the recurrences. Summing over \(k\ge0\) yields the first-moment asymptotics
\[
m_\lambda:=\frac1{2^\lambda}\sum_{2^\lambda\le t<2^{\lambda+1}} c_t
=\frac12+\frac1{2\sqrt{\pi\lambda}}+O(1/\lambda),
\]
and similarly
\[
\tilde m_\lambda=\frac12-\frac1{2\sqrt{\pi\lambda}}+O(1/\lambda).
\]

The second moment is handled through a trivariate generating function. With
\[
a_{\lambda,k,\ell}=4^\lambda\sum_{2^\lambda\le t<2^{\lambda+1}}
\delta(\lambda+1-k,t)\,\delta(\lambda+1-\ell,t),
\]
one defines
\[
A(x,y,z)=\sum_{\lambda,k,\ell\ge0}a_{\lambda,k,\ell}x^\lambda y^k z^\ell,
\]
and the recurrences imply that \(A(x,y,z)\) is an explicit rational function. The mean of \(c_t^2\) over the same dyadic block becomes a diagonal coefficient of
\[
F(x,y,z)=\frac{A(x,y,z)}{(1-y)(1-z)}.
\]
Multivariate complex-analytic saddle-point methods in the sense of Pemantle–Wilson, or alternatively a guessed and checked univariate \(D\)-finite recurrence, then give
\[
\frac1{2^\lambda}\sum c_t^2
=
\frac14+\frac1{2\sqrt{\pi\lambda}}+O(1/\lambda),
\]
with an analogous formula for \(\tilde c_t^2\).

At that point Chebyshev’s inequality supplies concentration:
\[
\operatorname{Var}(c_t)=E(c_t^2)-[E(c_t)]^2=O(1/\lambda)
\]
while
\[
[E(c_t)-\tfrac12]^2=O(1/\lambda),
\]
so most \(t\in[2^\lambda,2^{\lambda+1})\) satisfy \(c_t>\frac12\). Summing over \(\lambda\) gives the density-one conclusion. The same paper also exhibited the explicit sequence
\[
t_j=\frac{4^j-1}{3}=(10)^{j-1}1_2,
\]
for which
\[
c_{t_j}=\frac12+\frac{\sqrt3}{4\sqrt{2\pi j}}+O(j^{-3/2}),
\]
hence \(c_{t_j}>\frac12\) for all \(j\) [1509.08623].

## 4. Quantitative lower bounds, almost-sure results, and numerical evidence

Later work refined the density-one picture. Spiegelhofer proved that for every \(\varepsilon>0\) there exists \(L=L(\varepsilon)\) such that whenever the binary expansion of \(t\) contains at least \(L\) blocks of consecutive \(1\)'s, one has
\[
c_t>\frac12-\varepsilon.
\]
As a corollary, there are constants \(C,\kappa>0\), depending on \(\varepsilon\), such that for all \(T\ge2\),
\[
\bigl|\{0\le t<T:c_t\le\tfrac12-\varepsilon\}\bigr|
<
C(\log T)^\kappa.
\]
This replaced earlier exceptional-set bounds of size \(\ll T/\log T\) by an effective \(O((\log T)^\kappa)\) estimate.

The proof uses the characteristic function
\[
Y_t(\theta)=\sum_j \delta(j,t)e^{2\pi i j\theta},
\]
together with the identity
\[
c_t=\frac12+\int_0^{1/2}\operatorname{Im}Y_t(\theta)\cot(\pi\theta)\,d\theta.
\]
A moment-generating function \(M_t(x)=\sum_j\delta(j,t)e^{jx}\) satisfies the same two-scale recurrences as the underlying distributions, and bounds for its moments depend polynomially on the number of blocks of \(1\)'s in the binary expansion of \(t\). A matrix-product representation for \(Y_t(\theta)\) yields the estimate
\[
|Y_t(\theta)|\le \exp(-c\,r\,\theta^2)
\]
when \(r\) is the number of such blocks and \(\theta\) is away from \(0\). Near \(\theta=0\), Taylor expansion and moment bounds control the integral [1910.13170].

The same paper records numerical evidence: a computer check for all \(t<2^{30}\) verified \(c_t>\frac12\), with minimal value approximately \(0.51639\) at
\[
t=(111101111011110111101111011111)_2
\]
and its digit-reverse [1910.13170].

A related almost-sure phenomenon arises from the Tu–Deng conjecture. For
\[
P_{t,k}=\frac{|S_{t,k}|}{2^k},
\]
Spiegelhofer and Wallner proved that for every \(\varepsilon>0\), the proportion of \(t\in\{1,\dots,2^k-2\}\) with
\[
\frac12-\varepsilon<P_{t,k}\le \frac12
\]
tends to \(1\) as \(k\to\infty\). They also proved that the Tu–Deng conjecture implies Cusick’s conjecture, thereby linking two distinct binary digit-sum problems [1707.07945].

## 5. Martingales, binary trees, and stopped random walks

A different structural approach reinterprets the measures \(\mu_t\) through non-autonomous dynamics on probability measures. Restricting first to odd indices, one introduces the maps
\[
L(t)=2t-1,\qquad R(t)=2t+1,
\]
and a partial order on odd integers generated by finite sequences of left and right steps. In this framework one works with centered measures \(P_t\) defined by
\[
\mu_t=\mu_1*P_t,\qquad P_1=\delta_0,
\]
together with the recurrences
\[
P_{2t}=P_t,\qquad
P_{2t+1}=\tfrac12\,\sigma_{-1}P_t+\tfrac12\,\sigma_{+1}P_{t+1}.
\]

For an infinite word \(w\in\{L,R\}^{\mathbb N}\), the pair \((P^L_{w(n)},P^R_{w(n)})\) evolves under the non-autonomous maps
\[
\Phi_L(\mu,\nu)=\Bigl(\tfrac12\,\sigma_{-1}\mu+\tfrac12\,\sigma_{+1}\nu,\nu\Bigr),
\qquad
\Phi_R(\mu,\nu)=\Bigl(\mu,\tfrac12\,\sigma_{-1}\mu+\tfrac12\,\sigma_{+1}\nu\Bigr).
\]
The same recursion can be encoded by planar binary trees \(T_v\). To each finite tree one associates a stopping time \(\tau_T\) for the simple symmetric random walk \(S_n\), and the law of the stopped position \(S_{\tau_T}\) is exactly \(P_v\).

This representation yields explicit structural information. If \(w\) contains \(|w|_L\) letters \(L\) and \(|w|_R\) letters \(R\), then
\[
\operatorname{supp}P_w=\{-(|w|_L+1),\dots,|w|_R+1\},
\]
and the two extreme atoms have masses \(2^{-(|w|_L+1)}\) and \(2^{-(|w|_R+1)}\). The symmetries
\[
P_w(d)=P_{\bar w}(-d),\qquad P_w(d)=P_{\overleftarrow w}(d)
\]
hold, where \(\bar w\) exchanges \(L\leftrightarrow R\) and \(\overleftarrow w\) reverses the word. Variance is monotone with respect to the partial order: \(s\preceq t\) implies \(\operatorname{Var}(P_s)\le \operatorname{Var}(P_t)\). Among words of fixed length, the constant words \(L^n,R^n\) minimize the variance, with
\[
\operatorname{Var}(P_{L^n})=2-2^{-n}\to2,
\]
while the alternating word \((LR)^n\) maximizes it, with
\[
\operatorname{Var}(P_{(LR)^n})\approx \frac{2n}{3}.
\]

This framework leads to a median-preserving formulation. The original conjecture follows once one proves
\[
P_t(\mathbb N)=P\{X_t\ge0\}\ge \frac12\qquad \forall t\in\mathbb N.
\]
A stronger conjecture proposed in this setting is
\[
P_{Lv}(\mathbb N)>\frac12,\qquad P_{Rv}(-\mathbb N)>\frac12
\]
for every finite word \(v\). Numerically, a search among odd \(t\le 12\,000\,001\) found that the global minima of \(t\mapsto P_t(\mathbb N)\) are exactly \(\frac12\), and that those minimizers always end in the letter \(R\); no counterexample to the stronger inequality \(P_{Lv}(\mathbb N)>\frac12\) was found up to that range [2605.08624].

## 6. Full proof and explicit lower bound

The full conjecture was proved by a first-exit argument based on exact deconvolution. The starting point is the identity
\[
\mu_t=\mu_1*P_t,
\]
where \(P_t\) is finitely supported and satisfies
\[
P_{2t}=P_t,\qquad
P_{2t+1}=\tfrac12\,\sigma_{-1}P_{t+1}+\tfrac12\,\sigma_1P_t,
\qquad P_1=\delta_0.
\]
The initial measure is explicit:
\[
\mu_1=\sum_{j\ge0}2^{-j-1}\delta_{1-j}.
\]

The deconvolution kernel \(P_t\) is realized as the law of a stopped simple random walk. For a finite binary word \(a\), let \(Sub(a)\) be the set of subsequences of \(a\), and let
\[
\tau_a=\min\{n\ge1:\xi_1\cdots\xi_n\notin Sub(a)\}
\]
for i.i.d. Bernoulli\((\tfrac12)\) variables \((\xi_i)\). If
\[
S_n=\sum_{i=1}^n(2\xi_i-1),
\]
then
\[
\mathcal L(S_{\tau_a})=P_{t_w}
\]
for the corresponding odd integer \(t_w=(1\,w\,1)_2\).

The proof then establishes a general median theorem for any finite subsequence order ideal \(D\subset\{0,1\}^*\): \(0\) is a two-sided median of the stopped walk,
\[
\mathbb P\{S_{\tau_D}\ge0\}\ge \frac12,\qquad
\mathbb P\{S_{\tau_D}\le0\}\ge \frac12.
\]
For principal ideals \(D=Sub(a)\), a marked-deletion versus marked-insertion argument yields a strict quantitative bias. If \(Z=|a|_0\), \(O=|a|_1\), and
\[
m=\min(Z,O+1),
\]
then
\[
\mathbb P\{S_{\tau_a}\ge0\}\ge \frac12+2^{-2m-1}.
\]

A final support-cutoff argument transfers this strict bias from the kernel \(P_t\) back to the original measures \(\mu_t\). The resulting theorem is
\[
c_t=
\lim_{N\to\infty}\frac1N\#\{0\le n<N:\ s_2(n+t)\ge s_2(n)\}
\ge \frac12+2^{-2s_2(t)-1}
\qquad (t\ge1).
\]
In particular, \(c_t>\frac12\) for every \(t\ge1\), proving Cusick’s conjecture. The same work emphasizes that the proof uses elementary properties of binary digits, simple random walks, and finite combinatorics of subsequences, and notes that analogous questions for bases \(q>2\) remain open [2606.23398].

Source: https://www.emergentmind.com/topics/cusick-conjecture