Uniform O(N) error bound for the Tchoukaillon array

Prove that the Tchoukaillon array entries satisfy the uniform sharpened asymptotic estimate $\sqrt{Ð§_{i,j}}=L_{i,j}+O(1)$, equivalently $Ð§_{i,j}=\frac{(\pi i+2j+2)^2}{4\pi}+O(N)$, where $N=i+j+1$ and $L_{i,j}=\frac{\pi i+2j+2}{2\sqrt\pi}$, thereby improving the proven $O(N^{4/3})$ error and reducing the contour width from $O(V^{1/6})$ to $O(1)$.

Background

The paper proves the uniform estimate $Ð§_{i,j}=\frac{(\pi i+2j+2)^2}{4\pi}+O(N^{4/3})$ , or equivalently a square-root error of $O(N^{1/3})$ . Numerical experiments over millions of cells indicate that the square-root deviation remains bounded, with observed extrema near the array's two edges and nearly constant behavior in the interior. This motivates the conjectured improvement to a uniform $O(1)$ square-root error, which would imply an $O(N)$ error for the entries themselves and an $O(1)$ boundary width for the level contours.

The authors explain that their one-cell estimate at a single level does not control correlations among the nearest-integer remainders $E_m,F_m$ , or equivalently cancellation among the floor residues in the trace recurrence. Establishing the conjectured bound therefore requires new control of these cross-level correlations. The problem is explicitly left unresolved.

References

This points to the sharper

\sqrt{Ð§{i,j}}=L{i,j}+O(1),\qquad\text{equivalently}\qquad Ð§_{i,j}=\frac{(\pi i+2j+2)^{2}{4\pi}+O(N),}

under which the contour width of Corollary~\ref{cor:contour} would be $O(1)$ rather than $O(V^{1/6})$.

There is a precedent for this last step. On the row edge $i=0$ the sequence is the Tchoukaillon numbers, for which Erd\H os and Jabotinsky proved $Ð§_{0,j}=(j+1)^{2/\pi+O(j^{4/3})$} and conjectured the $O(j)$ later established by Broline--Loeb eq:brolineloeb. The exponent improvement $\tfrac43\to1$ we conjecture in the interior is exactly the Erd\H os--Jabotinsky improvement on the edge, now sought in every direction. Our theorem falls short of eq:conjecture because its bound is a pointwise one-cell estimate read at a single level; closing the gap would require the correlations among the nearest-integer remainders $E_m,F_m$ across levels---equivalently, the cancellation in the floor residues $r^{X_k,r^Y_k$} of Lemma~\ref{lem:mass}---which the present argument does not control. We leave eq:conjecture open.

eq:brolineloeb:

$Ð§_{0,j}=Ð§_{j+1}=\tfrac{1}{\pi}\,(j+1)^2+O(j+1)$

eq:conjecture:

$\sqrt{Ð§_{i,j}}=L_{i,j}+O(1),\qquad\text{equivalently}\qquad Ð§_{i,j}=\frac{(\pi i+2j+2)^2}{4\pi}+O(N),$

— Asymptotics of the Tchoukaillon array and a conjecture of Beluhov (2608.17517 - Li, 18 Aug 2026) in Section 'Concluding remarks', subsection 'The finer error term', equation (\ref{eq:conjecture})

Uniform O(N) error bound for the Tchoukaillon array

Background

References

Related Problems