The equality cases $P_t(\mathbb{N})=\tfrac12$ for the deconvolved sum-of-digits measures

Dawid Tar{\l}owski

The equality cases $P_t(\mathbb{N})=\tfrac12$ for the deconvolved sum-of-digits measures

T0 review · 0 major / 4 minor · reviewed 2026-08-28 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The deconvolved sum-of-digits measure assigns exactly half of its mass to the nonnegative integers precisely when the inner binary word is saturated.

desk verdict Settles Cheng's saturation problem with a clean characterization; the proof checks out, with only a couple of presentational gaps. read the letter →

arxiv 2608.24948 v1 pith:3YJIIU2Q submitted 2026-08-24 math.NT math.COmath.PR

Dawid Tar{l}owski This is my paper · ORCID

classification math.NTmath.COmath.PR MSC 11A6360G4005A0505A20

keywords sumofdigitsmedianpropertybinarytreesstoppedrandomwalkssubwordordersaturatedwordsasymptoticdensitiesexpansion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper solves the saturation problem for the deconvolved sum-of-digits measures: it gives a complete, finite characterization of the odd integers $t$ for which the probability measure $P_t$ assigns exactly mass $1/2$ to the nonnegative integers. Writing the odd integer as $t=(1w1)_2$, the paper proves that $P_t(\mathbb{N})=1/2$ if and only if the inner binary word $w$ is saturated, meaning that in the block decomposition $w=1^{a_0}01^{a_1}0\cdots01^{a_k}$ with $k$ zeros, every block of ones, including the first and last, has length at least $k$. It also proves that every non-saturated $w$ gives a strict excess, with $P_w(\mathbb{N})\ge 1/2+2^{-2\min(\ell_0(w),\ell_1(w)+1)-1}$. The interest is that the median inequality $P_t(\mathbb{N})\ge1/2$ lies behind the long-standing sum-of-digits median conjecture; identifying the equality cases turns that inequality into a precise, finitely checkable combinatorial statement and shows that equality is rare.

What carries the argument

The load-bearing object is the tree representation of the measures: $P_w$ is the law of a fair random walk stopped when it first leaves the internal nodes of a recursively grown binary tree $T_w$, and the internal node language of $T_w$ is the set of scattered subwords, obtained by deleting some digits, of the reversed word $\overleftarrow{w}$. The paper writes the excess $P_w(\mathbb{N})-1/2$ as a finite sum $\sum_{m\ge1}2^{-2m-1}\Delta_m^-(w)$, where $\Delta_m^-(w)=2|B_{2m-1}^{-1}(w)|-|B_{2m}^0(w)|$ and $B_n^x(w)$ is the number of subwords of $w$ of length $n$ and balance $x$. An insertion-counting identity then yields the key equivalence: $P_w(\mathbb{N})=1/2$ iff every subword $v$ of $w$ with $\ell_0(v)=\ell_1(v)+1$ has the property that inserting a single digit $1$ into any slot still gives a subword of $w$.

What would settle it

Evaluate the finite leaf sum of Corollary 4.8 for a saturated word with two zeros, for instance $w=11011011$: the theorem predicts $P_t(\mathbb{N})=1/2$ exactly, so any deviation falsifies it. Conversely, evaluate the same finite sum for the non-saturated word $w=1001$, whose middle block of ones between the two zeros has length $0<2$; the theorem predicts a value strictly larger than $1/2$.

Watch

Extended reading notes

Core claim

Theorem 1.2 is the paper's central claim: for $w\in\{0,1\}^*$ and $t=(1w1)_2$, $P_t(\mathbb{N})=1/2$ if and only if $w$ is saturated. Saturation means that when $w$ is written as $1^{a_0}01^{a_1}0\cdots01^{a_k}$ with exactly $k$ zeros, every $a_i$ satisfies $a_i\ge k$; equivalently, every zero is isolated and every maximal run of ones, including the first and last runs, has length at least the number of zeros. The proof establishes both directions: for saturated $w$, every subword $v$ with one more zero than ones remains a subword after inserting a single digit $1$ in any slot, which forces all correction terms in the formula for $P_w(\mathbb{N})$ to vanish; for non-saturated $w$, the paper constructs a short deficient subword whose forbidden insertion contributes a positive correction, giving the explicit lower bound.

Load-bearing premise

The characterization rests on the previously established identification of $P_w$ with the law of a stopped random walk on a recursively generated binary tree, together with an unproved combinatorial observation about block profiles that is used in both directions of the proof; if either of these failed, the saturation criterion would not control the measures.

Editorial extensions

If this is right

The equality set $\{t:P_t(\mathbb{N})=1/2\}$ has natural density zero, since the saturated words are exponentially sparse; the shortest saturated word with $k$ zeros has length $k(k+2)$.
For every non-saturated $w$, the strict inequality $P_w(\mathbb{N})>1/2$ holds with the explicit gap $2^{-2\min(\ell_0(w),\ell_1(w)+1)-1}$, extending the earlier lower bound beyond the $0$-initial case.
Corollary 1.3 gives an exact count of saturated words of each length, $E(\ell)=\sum_k\binom{\ell-k(k+1)}{k}$, so the equality cases are fully enumerated, not merely described qualitatively.
The saturation criterion is checkable in linear time from the run-length profile of $w$, so deciding whether a given odd integer is an equality case requires no numerical evaluation of the measure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper does not state the negative-side analogue, but the complement symmetry of the subword counts implies that $P_w(-\mathbb{N})=1/2$ should hold exactly when the complemented word $\overline{w}$ is saturated; this is a direct corollary of the same machinery.
The insertion-counting proof suggests a general rule: for any tree-defined stopped random-walk measure, the median defect is controlled by the shortest subword with one excess zero, so the exponential constant in lower bounds is set by a minimal deficient subword rather than by the whole tree geometry.
Because saturated words are exponentially rare, the strict median inequality $P_t(\mathbb{N})>1/2$ holds for a set of parameters of density one; the paper notes the zero-density conclusion but does not phrase it this way.

Share X Bluesky LinkedIn Reddit HN

Formalized claims in Lean

Claim #1: Theorem 1.2 is the paper's central claim: for $w\in\{0,1\}^*$ and $t=(1w1)_2$, $P_t(\mathbb{N})=1/2$ if and only if $w$ is saturated. Saturation means that when $w$ is written as $1^{a_0}01^{a_1}0\cdots01^{a_k}$ with exactly $k$ zeros, every $a_i$ satisfies $a_i\ge k$; equivalently, every zero is isolated and every maximal run of ones, including the first and last runs, has length at least the num

/-- @claim 1 Theorem 1.2 is the paper's central claim: for $w\in\{0,1\}^*$ and $t=(1w1)_2$, $P_t(\mathbb{N})=1/2$ if and only if $w$ is saturated. Saturation means that when $w$ is written as $1^{a_0}01^{a_1}0\cdots01^{a_k}$ with exactly $k$ zeros, every $a_i$ satisfies $a_i\ge k$; equivalently, every zero is isolated and every maximal run of ones, including the first and last runs, has length at least the num -/
def central_claim : Prop :=

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

This paper settles the saturation problem Cheng left open: for odd t=(1w1)_2, P_t(N)=1/2 iff the digit word w is saturated (with k zeros, every block of ones has length at least k). The characterization is genuinely new—[9] only had numerics, [2] left it open—and the proof is clean. The main counting identity (Prop 5.1) is a nice double-count of marked ones vs insertion slots; Cor 5.5 reduces equality to a subword-insertion property, and Observation 6.1 plus Lemma 6.2 turn that into the block-profile criterion. I checked the small cases and the count for the shortest saturated words; they match. The lower bound extension (Cor 6.3) also works: for non-saturated w it yields exactly Cheng's bound with the same exponent.

The soft spots are minor. Observation 6.1 is asserted without proof even though it is load-bearing in both directions of Theorem 1.2. It is an easy exercise—part (1) is a greedy embedding, part (2) is forced because equal numbers of zeros mean each zero of the subword maps to the corresponding zero of w—but a referee should ask for it to be written out. The other reliance is on [9]'s tree representation P_w = L(S_{\tau_w}) and the recurrences (2.1), (2.8). That is prior cited work, and Section 4 re-derives the one key ingredient (node language = reversed subwords) from scratch, so the dependence is transparent and not circular. The asymptotic-density-zero claim is tossed off as 'easy to note'; it follows from the isolated-zeros bound, but a line or two would be welcome. None of this threatens the main theorem.

Who is this for? Specialists in sum-of-digits measures and Cusick-type problems. It is not a paradigm shift, but it solves a concrete open question and gives a formula for the equality set. I would cite it if I worked in that area, and it deserves a real referee rather than a desk reject. My recommendation: send to review, and ask for a short proof of Observation 6.1 and a citation-level check of the imported tree representation.

Referee Report

0 major / 4 minor

Summary. The paper studies the family of finitely supported measures P_t obtained by deconvolving the sum-of-digits correlation measures μ_t. For an odd integer t=(1w1)_2, the main result (Theorem 1.2) is that P_t(N)=1/2 if and only if the inner word w is saturated, i.e. in the block decomposition w=1^{a_0}0...01^{a_k} with k zeros every block length satisfies a_i≥k. The proof passes through the tree representation of [9] and the subword description of node languages from [2], reduces P_w(N) to an insertion-counting series, and shows that the deficit terms vanish exactly in the saturated case. The paper also extends Cheng's lower bound to all non-saturated w and gives an explicit enumeration formula for saturated words.

Significance. The result resolves completely the saturation problem left open in [2] and previously studied only numerically. The characterization is clean and testable, and the proof is elementary and mostly self-contained after Section 4, with explicit verifiable identities (Proposition 5.1, Corollary 5.3, Theorem 5.8). The strengthened lower bound for non-saturated words is a quantitative bonus. I found no load-bearing mathematical error; the main theorem is sound, assuming the cited tree representation of [9] is correct. The only in-paper gap is the unproved Observation 6.1, which I verified; supplying its short proof is a minor revision.

minor comments (4)

[Section 6, Observation 6.1] This observation is asserted with the remark that the proof is an easy exercise, but it is used in both directions of Theorem 1.2: part (1) in the sufficiency argument and part (2) in Lemma 6.2. I verified the statement: part (1) follows by greedily embedding the blocks of u into the blocks of w, each of length at least k, and part (2) follows because an embedding of a word with exactly k zeros must send the j-th zero of u to the j-th zero of w. Please include this short proof (or at least a detailed sketch) so that the main theorem is fully supported in the manuscript itself.
[Section 5, proof of Theorem 5.2] In the proof of the second inequality in (5.4), the complement map sends B^x_n(w) to B^{-x}_n(\bar w), so the displayed chain should involve the complement of w rather than w itself; as written, the notation is confusing. Please correct this.
[Section 6, first line] There is a typo: 'We start with twe following' should be 'We start with the following'.
[Section 5, Corollary 5.9] The statement of Corollary 5.9 uses P_t(N) although the hypothesis is expressed for w; for consistency with the rest of the paper, either write P_w(N) or recall the identification t=(1w1)_2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the saturation characterization is derived from a self-contained insertion-counting argument; the cited tree framework is scaffolding, not a forced input.

full rationale

The derivation of Theorem 1.2 is self-contained after the imported tree/subword setup. The paper does not fit a parameter and then call it a prediction: P_w(N) is expressed by the telescoping identity (3.5)/(4.12), with Delta^-_m(w) a genuine count of subwords, and Corollary 5.5 reduces the equality P_w(N)=1/2 to an insertion property. Theorem 1.2 then follows from Observation 6.1, an elementary subword-block lemma that is asserted without proof (Section 6, 'the proof is an easy exercise') but is independent of P_t and is easily verified: saturated block profiles permit embedding of any word with at most k zeros and k ones, and with equal zero counts the block-wise comparison c_j <= a_j is necessary and sufficient. Lemma 6.2 and Corollary 6.3 are derived from this lemma and Corollary 5.9, not assumed. The only same-author citations are [9] for the recurrences (2.1), the tree growth (2.3), and the identification P_w = L(S_{tau_w}) (2.8); these are prior structural results with stated assumptions that do not include the saturation criterion, and the combinatorial Proposition 4.4 from [2] is re-proved in Section 4. Thus the claimed characterization is not equivalent by construction to any input; no circular step is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. The central claim depends on the prior structural encoding of P_t via trees, imported from [9], plus standard subword combinatorics that the paper proves or cites.

assumptions (3)

domain assumption P_1=delta_0, P_{2t}=P_t, P_{2t+1}=Phi(P_{t+1},P_t), and mu_t=mu_1*P_t (eqs. (2.1), (2.2)).
Imported from [9, Corollary 7]; these recurrences define the measures whose median property is studied.
domain assumption For the tree family defined by (2.3), the law of the stopped fair random walk equals P_w (eq. (2.8), [9, Theorem 11]).
Imported from [9]; the entire saturation theorem is stated for these tree measures.
standard math Cylinder probabilities and fair increments: P([u])=2^{-ell(u)} and S_n=b(u) on [u].
Standard product-measure facts used in Sections 2 and 3.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The equality cases $P_t(\mathbb{N})=\tfrac12$ for the deconvolved sum-of-digits measures." pith.science (2026). https://pith.science/paper/3YJIIU2Q

@misc{pith2026260824948,
  author       = {Pith},
  title        = {Pith review of: The equality cases $P_t(\mathbbN)=\tfrac12$ for the deconvolved sum-of-digits measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3YJIIU2Q}},
  note         = {Machine review of arXiv:2608.24948}
}

abstract

Let $s(n)$ denote the number of ones in the binary expansion of an integer $n\in\mathbb{N}$, and let $\mu_t$ be the probability measure on $\mathbb{Z}$ defined by the asymptotic densities of the level sets of the function $\mathbb{N}\ni n\mapsto s(n+t)-s(n)\in\mathbb{Z}$. Let $P_t$ be the family of finitely supported measures defined by the convolution $\mu_t=\mu_1*P_t$. Recently, Tarlowski (2026) has shown that the family $P_t$ may be represented as a recursively grown binary tree $T_t$, and that the Cusick's conjecture - $\mu_t(\mathbb{N})>\frac12$, $t\in\mathbb{N}$, - follows from the asymmetry property of the family $T_t$, which was posed there as an open problem. Next, Cheng (2026) has provided the combinatorial description of the family $T_t$ in the language of principal subsequence ideals, and proved both conjectures. Both of these problems are directly related to the problem of determining the zeros of the function $\mathbb{N}\ni t \mapsto P_t(\mathbb{N})-\frac12\in[0,\tfrac12]$, a problem left open by Cheng (2026) as a saturation problem, and previously analyzed only numerically. In this paper we solve this problem completely. Writing an odd integer $t\ge3$ as $t=(1\,w\,1)_2$ with $w\in\{0,1\}^{\star}$, we show that $P_t(\mathbb{N})=\frac12$ if and only if $w$ is \emph{saturated} in the following sense: in the block decomposition $w=1^{a_0}\,0\,1^{a_1}\,0\cdots0\,1^{a_k}$ with exactly $k$ zeros, every block of "1" satisfies $a_i\ge k$. Additionally, we show that the lower bound for $P_t(\mathbb{N})$ established by Cheng for $0$-initial words holds true for all non-saturated words.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 8 canonical work pages

[9]

On the sum-of-digits measures and Cusick's conjecture via stopped random walks

D. Tar lowski,On the sum-of-digits measures and Cusick’s conjecture via stopped random walks, arXiv:2605.08624, 2026. Instytut Matematyki, Wydzia l Matematyki i Informatyki, Uniwersytet Jagiello ´nski, ul. Lojasiewicza 6, 30-348 Krak´ow, Poland Email address:dawid.tarlowski@uj.edu.pl ; dawid.tarlowski@gmail.com

work page Pith review arXiv 2026
[2]

A first-exit proof of Cusick's sum-of-digits conjecture

K. Cheng,A first-exit proof of Cusick’s sum-of-digits conjecture, arXiv:2606.23398, 2026

work page Pith review arXiv 2026
[1]

somme des chiffres

J. B´ esineau,Ind´ ependance statistique d’ensembles li´ es ` a la fonction “somme des chiffres”, Acta Arith.20 (1972), 401–416

work page 1972
[3]

Drmota, M

M. Drmota, M. Kauers and L. Spiegelhofer,On a conjecture of Cusick concerning the sum of digits ofn andn+t, SIAM J. Discrete Math.30(2016), no. 2, 621–649

work page 2016
[4]

Emme and P

J. Emme and P. Hubert,Central limit theorem for probability measures defined by sum-of-digits function in base 2, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)19(2019), no. 2, 757–780

work page 2019
[5]

J. F. Morgenbesser and L. Spiegelhofer,A reverse order property of correlation measures of the sum-of- digits function, Integers12(2012), Paper No. A47, 5 pp

work page 2012
[6]

Sobolewski and L

B. Sobolewski and L. Spiegelhofer,Decomposing the sum-of-digits correlation measure, J. Number Theory 280(2026), 702–736

work page 2026
[7]

Spiegelhofer,A lower bound for Cusick’s conjecture on the digits ofn+t, Math

L. Spiegelhofer,A lower bound for Cusick’s conjecture on the digits ofn+t, Math. Proc. Cambridge Philos. Soc.172(2022), no. 1, 139–161

2022

Show all 9 references

[8]

Spiegelhofer and M

L. Spiegelhofer and M. Wallner,The binary digits ofn+t, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)24 (2023), no. 1, 1–31

2023

Pith tools

Reviewed August 28, 2026 · model on record in the stance chip above.

[1] [9]

On the sum-of-digits measures and Cusick's conjecture via stopped random walks

D. Tar lowski,On the sum-of-digits measures and Cusick’s conjecture via stopped random walks, arXiv:2605.08624, 2026. Instytut Matematyki, Wydzia l Matematyki i Informatyki, Uniwersytet Jagiello ´nski, ul. Lojasiewicza 6, 30-348 Krak´ow, Poland Email address:dawid.tarlowski@uj.edu.pl ; dawid.tarlowski@gmail.com

work page Pith review arXiv 2026

[2] [2]

A first-exit proof of Cusick's sum-of-digits conjecture

K. Cheng,A first-exit proof of Cusick’s sum-of-digits conjecture, arXiv:2606.23398, 2026

work page Pith review arXiv 2026

[3] [1]

somme des chiffres

J. B´ esineau,Ind´ ependance statistique d’ensembles li´ es ` a la fonction “somme des chiffres”, Acta Arith.20 (1972), 401–416

work page 1972

[4] [3]

Drmota, M

M. Drmota, M. Kauers and L. Spiegelhofer,On a conjecture of Cusick concerning the sum of digits ofn andn+t, SIAM J. Discrete Math.30(2016), no. 2, 621–649

work page 2016

[5] [4]

Emme and P

J. Emme and P. Hubert,Central limit theorem for probability measures defined by sum-of-digits function in base 2, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)19(2019), no. 2, 757–780

work page 2019

[6] [5]

J. F. Morgenbesser and L. Spiegelhofer,A reverse order property of correlation measures of the sum-of- digits function, Integers12(2012), Paper No. A47, 5 pp

work page 2012

[7] [6]

Sobolewski and L

B. Sobolewski and L. Spiegelhofer,Decomposing the sum-of-digits correlation measure, J. Number Theory 280(2026), 702–736

work page 2026

[8] [7]

Spiegelhofer,A lower bound for Cusick’s conjecture on the digits ofn+t, Math

L. Spiegelhofer,A lower bound for Cusick’s conjecture on the digits ofn+t, Math. Proc. Cambridge Philos. Soc.172(2022), no. 1, 139–161

2022