ChatBent Fuglede conjectured in 1974 that a geometric domain can support an orthogonal basis of exponential waves if and only if it can tile space; this mathematical proof confirms the generalized conjecture holds for a broad class of fractal Cantor-Moran measures.
In harmonic analysis, the fundamental question of Fourier duality asks: which geometric domains allow complex functions to be decomposed into orthogonal sets of pure harmonic waves?
Fuglede's conjecture posited a beautiful equivalence between spectrality (supporting an orthogonal Fourier basis) and tiling (covering space without gaps or overlaps). While the conjecture famously failed for general Euclidean dimensions, its behavior on singular fractal measures remained an open frontier.
This work establishes that the generalized Fuglede conjecture holds rigorously for a significant class of fractal Cantor-Moran measures, proving that spectral properties in these self-similar structures are intrinsically linked to admissible geometric translation tilings.
This mathematical breakthrough deepens our understanding of wave propagation in irregular fractal media, offering analytical foundations for quantum wave mechanics in disordered systems and fractal signal processing.
The generalized Fuglede’s conjecture holds for a class of Cantor–Moran measures
Suppose is a sequence of integers bigger than 1 and is a sequence of consecutive digit sets. Let be the Cantor-Moran measure defined by We prove that possesses an exponential orthonormal basis if and only if for some Borel probability measure . This theorem shows that the generalized Fuglede's conjecture is true for such Cantor-Moran measure. An immediate consequence of this result is the equivalence between the existence of an exponential orthonormal basis and the integral tiling of for .
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Calculating the number of ways to break large numbers into smaller pieces once required weeks of grueling manual additions; G. H. Hardy and Srinivasa Ramanujan invented the Circle Method to calculate exact partitions in fractions of a second. Published in 1918 from their legendary Cambridge collaboration, this masterwork founded analytic additive number theory and is the mathematical engine calculating black hole entropy in modern quantum string theory.
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