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Decomposing the sum-of-digits correlation measure

Bartosz Sobolewski, Lukas Spiegelhofer

TL;DR

This work develops a structural decomposition for the binary sum-of-digits correlation function $\gamma_t(\vartheta)$ by embedding the problem in an extended binary-expansion framework and encoding carries via matrix products. The main result expresses the extended $\gamma_{\mathbf t}$ as a finite sum of components with universal weights, clarifying how block lengths and their arrangement govern the correlations, and enabling explicit computations of the induced densities $c_{\mathbf t}$. By specializing to cases with at most two $1$-blocks and by analyzing Thue–Morse correlations, the paper derives concrete lower bounds for Cusick's conjecture and provides exact expressions for key densities in canonical configurations. The results shed light on the nonlocal carry effects that drive the central tendencies in the distribution of $\mathsf s(n+t)-\mathsf s(n)$ and propose concrete conjectures for extremal configurations and block-append operations, suggesting new avenues toward the unresolved Cusick conjecture.

Abstract

Let $s(n)$ denote the number of ones in the binary expansion of the nonnegative integer $n$. How does $s$ behave under addition of a constant $t$? In order to study the differences \[s(n+t)-s(n),\] for all $n\ge0$, we consider the associated characteristic function $γ_t$. Our main theorem is a structural result on the decomposition of $γ_t$ into a sum of \emph{components}. We also study in detail the case that $t$ contains at most two blocks of consecutive $1$s. The results in this paper are motivated by \emph{Cusick's conjecture} on the sum-of-digits function. This conjecture is concerned with the \emph{central tendency} of the corresponding probability distributions, and is still unsolved.

Decomposing the sum-of-digits correlation measure

TL;DR

This work develops a structural decomposition for the binary sum-of-digits correlation function by embedding the problem in an extended binary-expansion framework and encoding carries via matrix products. The main result expresses the extended as a finite sum of components with universal weights, clarifying how block lengths and their arrangement govern the correlations, and enabling explicit computations of the induced densities . By specializing to cases with at most two -blocks and by analyzing Thue–Morse correlations, the paper derives concrete lower bounds for Cusick's conjecture and provides exact expressions for key densities in canonical configurations. The results shed light on the nonlocal carry effects that drive the central tendencies in the distribution of and propose concrete conjectures for extremal configurations and block-append operations, suggesting new avenues toward the unresolved Cusick conjecture.

Abstract

Let denote the number of ones in the binary expansion of the nonnegative integer . How does behave under addition of a constant ? In order to study the differences for all , we consider the associated characteristic function . Our main theorem is a structural result on the decomposition of into a sum of \emph{components}. We also study in detail the case that contains at most two blocks of consecutive s. The results in this paper are motivated by \emph{Cusick's conjecture} on the sum-of-digits function. This conjecture is concerned with the \emph{central tendency} of the corresponding probability distributions, and is still unsolved.

Paper Structure

This paper contains 20 sections, 18 theorems, 160 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. An extension to infinite blocks
  4. Properties of the extension
  5. Evaluating the formal power series $\gamma$
  6. Generalized densities for Cusick's conjecture
  7. Main theorems
  8. Remarks and corollaries
  9. Dissection by infinite blocks
  10. The "reverse order property"
  11. Correlations of the Thue--Morse sequence
  12. Proof of Theorem \ref{['thm_components']}
  13. Explicit computation of $c_\mathbf t$
  14. Evaluating the base cases
  15. The case $N=1$
  16. ...and 5 more sections

Key Result

Theorem A

There exists an effective constant $N_0$ with the following property. If the natural number $t$ has at least $N_0$ maximal blocks of $\mathtt 1$s in its binary expansion, then $c_t > 1/2$.

Figures (3)

  • Figure 1: Real part of $(-1)^n\mathfrak g_{3,n}(\vartheta)\mathfrak C_{3,n}(\vartheta)$ for $t=153$ and $n\in\{0,1,2,3\}$.
  • Figure 2: Some values $-3\gamma_t$, where $-3\gamma_{19}$ and $-3\gamma_{25}$ are boxed.
  • Figure 3: An array containing $\mathfrak B_0(0),\ldots,\mathfrak B_5(0)$ as its diagonal

Theorems & Definitions (42)

  • Definition 1
  • Conjecture 1
  • Theorem A
  • Example
  • Remark 1
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Theorem 4
  • ...and 32 more