OP-01035 Partially resolved

Independent summands of a uniformly distributed sum

Let $X$ and $Y$ be independent random variables with finite support in $\mathbb{Z}$, such that $X + Y$ is uniformly distributed on its range. Such pairs exist with $X$ and $Y$ uniform: for instance $X$ uniform on $\{n, 2n, \dots, (n-1)n\}$ and $Y$ uniform on $\{0, 1, \dots, n-1\}$.

Problem. Are $X$ and $Y$ necessarily uniformly distributed on their ranges?

Source: Asked by E. Amiot on Mathematics Stack Exchange (2019), then discussed on MathOverflow; in polynomial form the question goes back to G. Letac (around 1969). Listed as Problem 28 in B. Green, 100 open problems (version of January 2026).

Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Classification

Primary: PR — Probability
Secondary: CO
Keywords: 0-1 polynomials, sums of independent random variables, uniform distribution

References

@misc{amiot2019mse,
  author = {Amiot, Emmanuel},
  title = {The coefficients of a product of monic polynomials are 0 and 1; if the polynomials' coefficients are non-negative, must they also be 0 and 1?},
  howpublished = {Mathematics Stack Exchange, question 3325163, \url{https://math.stackexchange.com/q/3325163}},
  year = {2019}
}

@misc{mo339137,
  author = {{Various authors}},
  title = {Why do polynomials with coefficients 0,1 like to have only factors with 0,1 coefficients?},
  howpublished = {MathOverflow, question 339137, \url{https://mathoverflow.net/q/339137}},
  year = {2019}
}

@misc{ghidelli2022unfair,
  author = {Ghidelli, Luca},
  title = {Progress on the unfair 0-1-polynomials conjecture using linear recurrences and numerical analysis},
  year = {2022},
  eprint = {2209.09843},
  archivePrefix = {arXiv}
}

@article{hare2025unfair,
  author = {Hare, Kevin G.},
  title = {Computational progress on the unfair 0-1 polynomial conjecture},
  journal = {Experimental Mathematics},
  pages = {1--9},
  year = {2025},
  doi = {10.1080/10586458.2025.2527786}
}

@misc{dvorsky2026unfair,
  author = {Dvorsky, Alexander},
  title = {The unfair 0-1 polynomial problem and high-degree trinomials},
  year = {2026},
  eprint = {2608.19173},
  archivePrefix = {arXiv}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 28; version of January 2026, most recent update December 2025},
  year = {2026}
}

Resolution claims

The Unfair 0-1 Polynomial Problem and High-Degree Trinomials v1 arXiv ↗ AI-Verified

Alexander Dvorsky

The unfair $0$--$1$ polynomial conjecture asks whether a factorization \[C(x)=A(x)B(x),\] with $A$ and $B$ monic and having nonnegative real coefficients, and $C$ a polynomial with all the coefficients 0 and 1, must already be a factorization into $0$--$1$ polynomials. Let $k$ be odd and $0<a<1$. We study the possibility that \[1+a x^2+x^k\] divides a $0$--$1$ polynomial with a nonzero cofactor …

Partial resolution · not verified · fidelity not established · posted on arXiv 1 Sep 2026

Status history

When	Change	By	Reason
2026-10-29	Open → Partially resolved	The registry's curators	The Unfair 0-1 Polynomial Problem and High-Degree Trinomials v1 recorded from arXiv:2608.19173v2.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-01035, "Independent summands of a uniformly distributed sum", clopen.solutions.