The central finding is an exact match. For the nearest-neighbor lattice inequality in three dimensions, the sharp coefficient is 1/4, the same coefficient that appears in the continuous Hardy inequality. The theorem establishes that inequality for every finitely supported complex-valued function on Z³. The document is arXiv:2608.25262v1, dated 26 Aug 2026.
The result answers a tightly framed mathematical question: does moving from the continuous setting to a grid alter the coefficient 1/4? The work considers every finitely supported function u: Z³ → C, rather than an empirical sample. Its inequality compares the discrete energy E₃(u) with mass weighted by an inverse-square term. The question is whether that comparison can hold uniformly at the continuous coefficient.
How the proof builds the bound
The proof is built around an edge field, a positive factor attached to a pair of neighboring points. For an edge from x to y, it uses a_xy = (5|x|² + 3|y|² + 8)/(3|x|² + 5|y|² + 8). Reversing the edge changes the factor to its reciprocal, so a_yx = a_xy⁻¹. The argument then completes squares one edge at a time and uses concavity of the resulting vertex weight.
That construction produces more than the bare Hardy estimate. The exact edge decomposition gives E₃(u) ≥ the sum over x in Z³ of W(x)|u(x)|², with W a vertex weight. At the origin, W(0) = 12/13. For every nonzero x, the weight is at least 1/(4|x|²) plus the correction ρ(|x|²), and equality occurs only on one of the three coordinate axes.
The correction is explicit rather than an unspecified error term. For n ≥ 1, the paper defines ρ(n) = (928n² + 5356n - 2197) / (4n(8n + 13)(64n² + 108n + 169)). In the improved inequality, the right-hand side is strictly positive whenever u is nonzero.
Sharp, but only as a limit
Here, sharp has a precise meaning: 1/4 is the largest coefficient that can hold across the stated class. The paper supports that claim with a sequence of admissible functions whose Rayleigh quotients, the ratios used to test the bound, tend to 1/4. The result is therefore about the best uniform constant, not about one specially chosen function.
The sequence is obtained by sampling continuous functions on scaled lattices. In the paper’s notation, the samples take the form u_{L,N}(x) = φ_L(x/N). The construction lets the lattice quotients approach the continuous target without producing a function that reaches it.
That distinction is central to the theorem. No nonzero finitely supported function attains the coefficient 1/4. The improved theorem also says its right-hand side is strictly positive whenever u is nonzero, while the sequence can come arbitrarily close in the quotient. In mathematical terms, the optimum is nonattained.
Where the result stops
The conclusion has a defined boundary. It addresses the three-dimensional lattice and the inverse-square weight used in this construction. It does not resolve exact pure-power constants in fixed dimensions d ≥ 4 or nonlinear discrete p-energies, both of which the paper leaves open.
Paper data and sources
Original title: The sharp discrete Hardy inequality on $\Z^3$
Authors: Natanael Alpay
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text