Geometric Moments and Rational Lambert Values: Proofs and Further Results

Why the choice of family matters.

The converse can start from an arbitrary small integer form and build the polynomial rows afterwards. Pigeonhole approximation supplies the small form; Bézout’s identity lifts its coordinates to polynomial evaluations. Thus the degree and residue conditions do not constrain how the approximation was obtained. Let $\xi$ be any irrational real number. Comparing the fractional parts of $0,\xi ,\dots ,(n+1)\xi$ in $n+1$ equal half-open intervals gives integers $A\ge 1$ and $B$ with

Set $w=n^2$, $W_n=R_n=S_n=w$, $M_n=6w$, and $D=6^w$. The three integer rows

are primitive and sum to $(DA,DB)$. Pad them to $M_n$ rows with $(1,0)$, and use the coefficient vector $(1,1,1,0,\dots ,0)$.

To realise every coordinate as a polynomial evaluation of degree at most $w$, choose integers $u,v$ such that $u{2}^w+v{3}^w=1$. For any integer $h$, the polynomial $h(u+v{X}^w)$ satisfies $H_w(h(u+v{X}^w))=h$. Lift all row coordinates this way. The combined polynomial pair is nonzero because its first evaluation is $DA\ne 0$. Its two evaluations are divisible by $3^w{2}^w$, and its divided remainder is $A\xi -B$. All the conditions of Problem 10.1, with $F(3/2)$ replaced by $\xi$, hold with $C=6$.

Even a coefficient-height bound of the form $\exp (O_{\xi }(n^2))$ would not exclude these lifts. Choose $0\le v<2^w$, which gives $|u|<3^w$. Since $|B|\le (n+1)|\xi |+1$, every lifted coefficient has absolute value at most $(|\xi |+2)(n+1){18}^w$. Together with the preceding rationality contradiction, this proves that the unrestricted construction problem for a real target $\xi$ is equivalent to the irrationality of $\xi$. It does not establish either assertion for $F(3/2)$.

The short paper refers to this formulation and to its equivalence with irrationality. To pose a more specific question, one must replace the free choice of polynomial pairs by an actual set given by coefficient formulas, admissible parameters and permitted normalisations. The restriction must exclude the freedom to choose the rows from an already good approximation. Merely giving the resulting family a name, an explicit formula or an exponential height bound does not do so: the lifts above have all three. For a fixed indexed family, the distinction between all sufficiently large indices and a sparse subsequence can matter. Arbitrary padding and reindexing remove it in the unrestricted problem.

Corollary 5.5 gives no improvement from integer rescaling and, under its endpoint hypotheses, no common factor $2$ or $3$ in the unscaled evaluations. It makes no general assertion about polynomial cancellation before specialisation, combinations of rows, determinant-specific divisibility or other approximation families.

A prescribed family must supply nonvanishing and decay for the same signed combinations. A residue collision alone supplies neither; separate examples of the two properties would not suffice either. A coefficient-height estimate is needed when the approximation argument calls for it, not as an extra hypothesis of the elementary irrationality criterion above.