Hostname: page-component-5b777bbd6c-7mr9c Total loading time: 0 Render date: 2025-06-19T08:01:24.741Z Has data issue: false hasContentIssue false
  • English
  • Français

DIFFRACTION OF THE PRIMES AND OTHER SETS OF ZERO DENSITY

Part of: Optics, electromagnetic theory - General Elementary number theory

Published online by Cambridge University Press:  21 May 2025

ADAM HUMENIUK Open the ORCID record for ADAM HUMENIUK [Opens in a new window] ,
CHRISTOPHER RAMSEY Open the ORCID record for CHRISTOPHER RAMSEY [Opens in a new window]  and
NICOLAE STRUNGARU Open the ORCID record for NICOLAE STRUNGARU [Opens in a new window]
ADAM HUMENIUK
Affiliation:
Department of Mathematics and Computing, Mount Royal University, Calgary, Alberta, Canada e-mail: ahumeniuk@mtroyal.ca
CHRISTOPHER RAMSEY
Affiliation:
Department of Mathematics and Statistics, MacEwan University, Edmonton, Alberta, Canada e-mail: ramseyc5@macewan.ca
NICOLAE STRUNGARU*
Affiliation:
Department of Mathematics and Statistics, MacEwan University, Edmonton, Alberta, Canada, and Institute of Mathematics ‘Simon Stoilow’, Bucharest, Romania
*
e-mail: strungarun@macewan.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we show that the diffraction of the primes is absolutely continuous, showing no bright spots (Bragg peaks). We introduce the notion of counting diffraction, extending the classical notion of (density) diffraction to sets of density zero. We develop the counting diffraction theory and give many examples of sets of zero density of all possible spectral types.

Keywords

diffractionautocorrelationprimes

MSC classification

Primary: 78A45: Diffraction, scattering
Secondary: 11A41: Primes
Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 44
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Communicated by Michael Coons

The first author was partially supported by the NSERC Discovery grant 2024-03883, the second author was supported by the NSERC Discovery grant 2019-05430, and the third author was supported by the NSERC Discovery grants 2020-00038 and 2024-04853.

References

Baake, M., ‘Diffraction of weighted lattice subsets’, Canad. Math. Bull. 45 (2002), 483–498.CrossRefGoogle Scholar
Baake, M. and Grimm, U., Aperiodic Order. Volume 1: A Mathematical Invitation, Encyclopedia of Mathematics and its Applications, 149 (Cambridge University Press, Cambridge, 2013).CrossRefGoogle Scholar
Baake, M., Huck, C. and Strungaru, N., ‘On weak model sets of extremal density’, Indag. Math. (N.S.) 28 (2017), 3–31.10.1016/j.indag.2016.11.002CrossRefGoogle Scholar
Baake, M. and Lenz, D., ‘Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra’, Ergodic Theory Dynam. Systems 24 (2004), 1867–1893.CrossRefGoogle Scholar
Baake, M. and Moody, R. V., ‘Weighted Dirac combs with pure point diffraction’, J. reine angew. Math. 573 (2004), 61–94.Google Scholar
Bateman, P. and Stemmler, R., ‘Waring’s problem for algebraic number fields and primes of the form $({p}^r-1)/({p}^d-1)$ ’, Illinois J. Math. 6 (1962), 142–156.CrossRefGoogle Scholar
Berg, C. and Forst, G., Potential Theory on Locally Compact Abelian Groups (Springer, Berlin, 1975).10.1007/978-3-642-66128-0CrossRefGoogle Scholar
Cowley, J. M., Diffraction Physics, 3rd edn (North-Holland, Amsterdam, 1995).Google Scholar
Dieudonné, J., Treatise on Analysis. Vol. 2 (Academic Press, New York, 1970).Google Scholar
Dworkin, S., ‘Spectral theory and X-ray diffraction’, J. Math. Phys. 34 (1993), 2965–2967.10.1063/1.530108CrossRefGoogle Scholar
Gouéré, J.-B., ‘Diffraction and Palm measure of point processes’, C. R. Math. Acad. Sci. Paris 336 (2003), 57–62.CrossRefGoogle Scholar
Grimm, U. and Deng, X., ‘Some comments on pinwheel tilings and their diffraction’, J. Phys.: Conf. Ser. 284 (2011), Article no. 012032.Google Scholar
Hof, A., ‘On diffraction by aperiodic structures’, Comm. Math. Phys. 169 (1995), 25–43.CrossRefGoogle Scholar
Keller, G. and Richard, C., ‘Dynamics on the graph of the torus parametrisation’, Ergodic Theory Dynam. Systems 38 (2018), 1048–1085.CrossRefGoogle Scholar
Keller, G. and Richard, C., ‘Periods and factors of weak model sets’, Israel J. Math. 229 (2019), 85–132.CrossRefGoogle Scholar
Maynard, J., ‘Small gaps between primes’, Ann. of Math. (2) 181 (2015), 383–413.10.4007/annals.2015.181.1.7CrossRefGoogle Scholar
Montgomery, H. L. and Vaughan, R. C., ‘The large sieve’, Mathematika 20 (1973), 119–134.CrossRefGoogle Scholar
Moody, R. V. and Strungaru, N., ‘Almost periodic measures and their Fourier transforms’, in: Aperiodic Order. Volume 2: Crystallography and Almost Periodicity, Encyclopedia of Mathematics and its Applications, 166 (eds. Baake, M. and Grimm, U.) (Cambridge University Press, Cambridge, 2017), 173–270.10.1017/9781139033862.006CrossRefGoogle Scholar
Pedersen, G. K., Analysis Now (Springer, New York, 1989); revised printing 1995.CrossRefGoogle Scholar
Polymath, D. H. J., ‘Variants of the Selberg sieve, and bounded intervals containing many primes’, Res. Math. Sci. 1 (2014), 83 pages.CrossRefGoogle Scholar
Richard, C. and Strungaru, N., ‘Fourier analysis of measures with lattice support’, in preparation.Google Scholar
Rudin, W., Real and Complex Analysis (McGraw-Hill, New York, 1987).Google Scholar
Samuel, , ‘What’s the asymptotic distribution of ${p}^n$ (powers of primes)?’, Mathstackexchange, q/396665, 2016. https://math.stackexchange.com/q/396665.Google Scholar
Schlottmann, M., ‘Generalised model sets and dynamical systems’, in: Directions in Mathematical Quasicrystals, CRM Monograph Series, 13 (eds. Baake, M. and Moody, R. V.) (American Mathematical Society, Providence, RI, 2000), 143–159.Google Scholar
Strungaru, N., ‘On the Fourier analysis of measures with Meyer set support’, J. Funct. Anal. 278 (2020), Article no. 108404.CrossRefGoogle Scholar
Torquato, S., Zhang, G. and de Courcy-Ireland, M., ‘Hidden multiscale order in the primes’, J. Phys. A 52 (2018), Article no. 135002.Google Scholar
Torquato, S., Zhang, G. and de Courcy-Ireland, M., ‘Uncovering multiscale order in the prime numbers via scattering’, J. Stat. Mech. 2018 (2018), Article no. 093401.CrossRefGoogle Scholar
Zhang, G., Martelli, F. and Torquato, S., ‘The structure factor of primes’, J. Phys. A 51 (2018), Article no. 115001.10.1088/1751-8121/aaa52aCrossRefGoogle Scholar
Zhang, Y., ‘Bounded gaps between primes’, Ann. of Math. (2) 179 (2014), 1121–1174.10.4007/annals.2014.179.3.7CrossRefGoogle Scholar