It has been known that every regular model set has a pure discrete spectrum, but the converse is not true in general. The relation between pure discrete spectrum and regular model sets is well studied in [2, 1, 4] in quite a general setting. When we restrict to substitution tilings, we can observe a more concrete relation between the two notions. It has been shown in [3] that pure discrete spectrum and regular model sets are equivalent in the setting of unimodular substitution tilings in $\mathbb{R}^{d}$.

In this talk, we show that a pure discrete spectrum and a regular model set are equivalent in the setting of substitution tilings in $\mathbb{R}^{d}$ dropping off the condition of the unimodularity.

References:

[1] Baake, M., Lenz, D., Moody, R.V., Characterization of model sets by dynamical

systems, Ergodic Theory Dynam. Systems 27 (2007), 341-382.

[2] Baake, M. and Moody, R. V. Weighted Dirac combs with pure point diffraction. J. Reine Angew. Math. 573 (2004), 6194.

[3] Lee, D.-I., Akiyama, S., Lee, J.-Y., Pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings, Acta Cryst. A 76 (2020), 600-610.

[4] Strungaru, N., Almost periodic pure point measures. Aperiodic order. Vol. 2, 271342, Encyclopedia Math. Appl., 166, Cambridge Univ. Press, Cambridge, 2017.