Sum of Cubes in F_3^n

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $V=\mathbb F_3^n$, where $\mathbb F_3=\mathbb Z/3\mathbb Z$, and regard $\{0,1\}^n$ as a subset of $V$. A cube means $L(\{0,1\}^n)$ for an invertible $\mathbb F_3$-linear map $L:V\to V$; no translation is part of this definition. Equivalently, for a basis $v_1,\ldots,v_n$ the cube consists of the subset sums $\sum_j\epsilon_jv_j$, $\epsilon_j\in\{0,1\}$. For subsets $B_1,\ldots,B_m\subset V$, their sum is $B_1+\cdots+B_m=\{b_1+\cdots+b_m:b_i\in B_i\}$.

Statement recorded in the supplied source.

For every integer $n\geq1$ and every choice of $100$ invertible linear maps $L_1,\ldots,L_{100}\in\operatorname{GL}_n(\mathbb F_3)$, is \[ \sum_{i=1}^{100}L_i(\{0,1\}^n)=\mathbb F_3^n? \] Equivalently, is each vector of $\mathbb F_3^n$ a $0/1$ linear combination of the multiset union of any $100$ bases? [S2]

Short English statement

Does adding any one hundred linear images of the binary cube in a vector space over the three-element field cover the entire vector space?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 157 (GREEN-069). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Yang Yu, The Permanent Rank of a Matrix (Part Three): Note on the Additive Basis Conjecture, arXiv:2510.01300v2 (2026), Introduction, Conjecture 1 and Theorem 3. The manuscript gives the stronger four-basis conclusion; since every cube contains zero, that conclusion implies the recorded one-hundred-cube statement. https://arxiv.org/abs/2510.01300.
• S3 Ben Green, 100 Open Problems, maintained author list, Problem 26 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.