Running least common multiples: distinct values and multiplicities

What the Hecke–Mahler comparison does and does not supply.

In the published Luca–Ouaknine–Worrell article [26], the same fixed integer coefficients are applied to shifts of one integer sequence. The resulting nonzero terms must have expanding gaps and uniform polynomial variation; the original sequence must also have polynomial growth. Theorem 6 then gives a fixed-base value criterion. Theorem 8 verifies that condition for $f(\lfloor m\vartheta +\rho \rfloor )$ when $f\in \mathbb{Z}[x]$ is nonconstant, $\vartheta ,\rho \in (0,1)$ and $\vartheta$ is irrational. These interval restrictions belong to that normalised combinatorial statement. Their main value theorem, Theorem 1, allows every real $\rho$ and every irrational real $\vartheta$, with any algebraic base $\beta$ satisfying $|\beta |>1$. Claim 10 uses a finite difference of order $\mathrm{deg}f+1$, which vanishes when no floor crossing occurs. This is a polynomial evaluated at a floor, not merely an integer sequence of polynomial growth. Our $m_a$ is instead a weighted lattice count. Proposition 12.2 separates floor crossings from its moving boundary, and Example 12.3 shows that the latter can survive a cubic difference even with no crossings.

There is a second distinction. In our displayed shift identity the coefficient of $m_{a+jr}$ is $15c_j{\gamma }_{a,jr}$, which can depend on $a$. For example, $P_1=2$, $P_2=12$ and $P_3=120$ give ${\gamma }_{0,1}=1/3$ but ${\gamma }_{1,1}=1/5$. These factors are required by the denominator chain. A support estimate for $D_{r,a}$ is therefore not by itself a verification of Definition 5 for the sequence $(m_a)$. No fixed-coefficient representation to which that theorem applies has been established here.

For a variable-denominator analogue, a useful hypothesis would be a fixed choice of coefficients and shifts $r_n\to \mathrm{\infty }$ for which ${\mathrm{\Delta }}_n=\{a\ge 0:D_{r_n,a}\ne 0\}$ is infinite and distinct members are at least $\eta r_n$ apart, for a fixed $\eta >0$. Even this would leave the uniform polynomial-variation requirement, for example

with $C,d$ independent of $n$, $a$ and $a^{\prime }$. This is a condition on the gaps between nonzero terms, not merely on their density: even a set of density zero can contain adjacent pairs. It also requires infinitely many nonzero terms for each selected shift. A small but nonzero term at every index would fail it, and an identically zero difference would also fail the infinitude requirement. The cited paper constructs suitable shifts for the polynomial-floor sequences just described; no such result is proved for our weighted counts. The bound $O((a+r+1{)}^2)$ controls absolute size: it depends on the location $a$ itself. The variation condition instead bounds a later nonzero term by the gap from an earlier one and the size of that earlier term, with constants uniform in the shift. The absolute-size bound alone gives no such comparison. The denominator chain itself is not a fixed-base power sequence. The next proposition shows what is lost in two direct fixed-base recodings. A value criterion for the original chain would need a separate proof, including non-cancellation and the Diophantine height estimates required by a Subspace-Theorem argument. These heights measure the numerators and denominators of the approximating algebraic quantities; they are unrelated to the running LCM $\mathrm{H}$. The finite set of prime divisors alone does not provide those estimates.