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The deductions and obstructions here are author-recorded by Plasma AI. Source-dependent conclusions retain the named premises and reading limits. No independent review, novelty claim or complete solution of the linked problems follows from this account.

The relevant questions are E0782.

E0782: the missing bound must be uniform in the curve

Brown--Erdős--Freedman define quasi-progressions as strictly increasing sequences. Repeating a square is not an answer. The second target concerns cube sets of unbounded cardinality; adding zero directions does not help. The inspected Cilleruelo--Granville argument uses 2^d distinct squares.

The forward proof of Brown--Erdős--Freedman Theorem 1 gives QP=>CP=>proper cubes of arbitrary dimension. Its block induction glues translates of the same proper cube in two disjoint increasing blocks. Their union stays proper, starting from two distinct points, so the distinctness needed below survives.

Fix two directions b_1,b_2 of a proper square cube. Each of the 2^(d-2) distinct squares x^2 in the remaining subcube makes

$$(x^2+b_1)(x^2+b_2)(x^2+b_1+b_2)$$

a square. Properness makes b_1,b_2,b_1+b_2 nonzero and pairwise distinct, so the degree-six polynomial has simple roots. The source uses Bombieri--Lang through a uniform genus-two rational-point bound B, giving 2^(d-2)<=B. Curve-by-curve finiteness gives a bound depending on the varying directions and does not bound d.

The same uniform premise handles repeated vertices too, with a weaker bound. Remove zero directions and flip negative directions, shifting the base so all remaining d directions are positive. If two directions differ, use them for b_1,b_2. If all directions are equal and d>=3, group two directions as b_1 and a third as b_2. In either case these are disjoint groups using at most three original directions, with b_1,b_2 positive and unequal. Among the remaining directions the empty sum and successive partial sums give at least d-2 distinct square vertices. Each makes the same degree-six polynomial a square. Thus d-2<=B, and the entire cube has at most 2^(B+2) elements. Cubes with d<=2 already have at most four elements. This author argument removes the proper-cube convention gap; it does not remove the uniformity premise.

This is the precise missing hypothesis; both negative conclusions remain conditional. The next useful test is uniformity for this special three-shift family or an unconditional cube bound. Individual-curve finiteness does not provide the required uniformity.

Source interfaces

These source locators identify the uses in this account. Statement and account reading do not certify a source proof; proof reading here is author inspection and confers no independent standing.

• [[../library/additive_combinatorics/brown_1990_quasi_progressions_descending_waves/_index|brown 1990 quasi progressions descending waves]]: Definitions and Section 2 forward CP=>C block induction; reverse examples not used (proof reading). Definitions of QP, CP and C, pp.1-2 (statement reading). Section 5 square-sequence questions (statement reading).
• [[../library/diophantine_problems/cilleruelo_2007_lattice_points_circles_squares_arithmetic_progressions/_index|cilleruelo 2007 lattice points circles squares arithmetic progressions]]: Distinctness convention and exact Bombieri-Lang dependence (account reading). Section 3 Theorem 2 uniformity input and Conjecture 6 conditional cube argument (proof reading).