{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T20:32:17Z","timestamp":1777667537668,"version":"3.51.4"},"reference-count":30,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T00:00:00Z","timestamp":1777334400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study the discrete-time amplitude-constrained additive white Gaussian noise (AWGN) channel from the perspective of near-optimal input distributions in the high-SNR, or equivalently large-amplitude, regime. While it is known that the capacity-achieving input is discrete with finitely many mass points, the precise scaling of its support size as a function of the amplitude constraint remains an open problem. In this work, we instead consider the minimal support size required to achieve capacity up to an \u03b5-gap. We introduce the quantity K\u03b5(A), defined as the smallest support size among discrete inputs supported on [\u2212A,A] that achieves mutual information within \u03b5 of capacity. We show that this relaxed formulation is significantly more tractable and admits sharp characterizations in several vanishing-gap regimes. In particular, for polynomially decaying gaps, \u03b5=A\u2212\u03b2 with \u03b2\u22651, we establish that K\u03b5(A)=\u0398(AlogA) as A\u2192\u221e. For exponentially small gaps, we obtain bounds of order between AlogA and A3\/2. Our approach combines approximation-theoretic bounds for Gaussian mixtures with information-theoretic control of entropy via \u03c72-divergence, together with a wrapping argument that relates the problem to approximating the uniform distribution on a circle. Beyond the technical results, our framework provides a conceptual explanation for the variety of scaling laws observed in prior numerical studies, suggesting that these may correspond to different regimes of \u03b5-optimality rather than intrinsic properties of the exact optimizer.<\/jats:p>","DOI":"10.3390\/e28050500","type":"journal-article","created":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:43:51Z","timestamp":1777448631000},"page":"500","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Support Size of \u03b5-Capacity-Achieving Inputs for the Amplitude-Constrained AWGN Channel"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4052-2092","authenticated-orcid":false,"given":"Luca","family":"Barletta","sequence":"first","affiliation":[{"name":"Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, 20133 Milano, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0625-5306","authenticated-orcid":false,"given":"Alex","family":"Dytso","sequence":"additional","affiliation":[{"name":"Qualcomm Flarion Technology, Inc., Bridgewater, NJ 08807, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,4,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2006","DOI":"10.1109\/TIT.2019.2948636","article-title":"The Capacity Achieving Distribution for the Amplitude Constrained Additive Gaussian Channel: An Upper Bound on the Number of Mass Points","volume":"66","author":"Dytso","year":"2020","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Dytso, A., Goldenbaum, M., Shamai, S., and Poor, H.V. (2017, January 4\u20138). Upper and lower bounds on the capacity of amplitude-constrained MIMO channels. Proceedings of the IEEE Global Communications Conference (GLOBECOM), Singapore.","DOI":"10.1109\/GLOCOM.2017.8254165"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Dytso, A., Goldenbaum, M., Poor, H.V., and Shamai, S. (2018, January 21\u201323). When Are Discrete Channel Inputs Optimal?\u2014Optimization Techniques and Some New Results. Proceedings of the Conference on Information Sciences and Systems, Princeton, NJ, USA.","DOI":"10.1109\/CISS.2018.8362306"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_5","unstructured":"Smith, J.G. (1969). On the Information Capacity of Peak and Average Power Constrained Gaussian Channels. [Ph.D. Dissertation, University of California]."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/S0019-9958(71)90346-9","article-title":"The information capacity of amplitude-and variance-constrained scalar Gaussian channels","volume":"18","author":"Smith","year":"1971","journal-title":"Inform. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1134\/S0032946010040022","article-title":"Transition points in the capacity-achieving distribution for the peak-power limited AWGN and free-space optical intensity channels","volume":"46","author":"Sharma","year":"2010","journal-title":"Probl. Inf. Transm."},{"key":"ref_8","unstructured":"Wang, H., Barletta, L., and Dytso, A. (2025). An Improved Lower Bound on Cardinality of Support of the Amplitude-Constrained AWGN Channel. arXiv."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1760","DOI":"10.1073\/pnas.1715306115","article-title":"Maximizing the information learned from finite data selects a simple model","volume":"115","author":"Mattingly","year":"2018","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1007\/s10955-019-02296-2","article-title":"A scaling law from discrete to continuous solutions of channel capacity problems in the low-noise limit","volume":"176","author":"Abbott","year":"2019","journal-title":"J. Stat. Phys."},{"key":"ref_11","unstructured":"Zhang, Z. (1994). Discrete Noninformative Priors. [Ph.D. Thesis, Yale University]."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1109\/TIT.1972.1054855","article-title":"Computation of channel capacity and rate-distortion functions","volume":"18","author":"Blahut","year":"1972","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1109\/TIT.1972.1054753","article-title":"An algorithm for computing the capacity of arbitrary discrete memoryless channels","volume":"18","author":"Arimoto","year":"1972","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_14","unstructured":"McKellips, A.L. (July, January 27). Simple tight bounds on capacity for the peak-limited discrete-time channel. Proceedings of the IEEE International Symposium on Information Theory, Chicago, IL, USA."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Dytso, A., Goldenbaum, M., Poor, H.V., and Shamai, S. (2019). Amplitude constrained MIMO channels: Properties of optimal input distributions and bounds on the capacity. Entropy, 21.","DOI":"10.3390\/e21020200"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4172","DOI":"10.1109\/TIT.2017.2692214","article-title":"Capacity Bounds for Discrete-Time, Amplitude-Constrained, Additive White Gaussian Noise Channels","volume":"63","author":"Thangaraj","year":"2017","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1290","DOI":"10.1109\/18.923716","article-title":"The capacity of discrete-time memoryless Rayleigh-fading channels","volume":"47","author":"Trott","year":"2001","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2257","DOI":"10.1109\/TIT.2004.834745","article-title":"On the capacity-achieving distribution of the discrete-time noncoherent and partially coherent AWGN channels","volume":"50","author":"Katz","year":"2004","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_19","first-page":"424","article-title":"Capacity of a pulse amplitude modulated direct detection photon channel","volume":"137","author":"Shamai","year":"1990","journal-title":"IEE Proc. I (Commun. Speech Vis.)"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"7050","DOI":"10.1109\/TIT.2021.3111836","article-title":"Properties of the Support of the Capacity-Achieving Distribution of the Amplitude-Constrained Poisson Noise Channel","volume":"67","author":"Dytso","year":"2021","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1178","DOI":"10.1109\/TIT.2017.2771815","article-title":"On properties of the support of capacity-achieving distributions for additive noise channel models with input cost constraints","volume":"64","author":"Fahs","year":"2017","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2773","DOI":"10.1109\/TIT.2004.836662","article-title":"On the discreteness of capacity-achieving distributions","volume":"50","author":"Tchamkerten","year":"2004","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2073","DOI":"10.1109\/TIT.2005.847707","article-title":"Capacity-achieving probability measure for conditionally Gaussian channels with bounded inputs","volume":"51","author":"Chan","year":"2005","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Abou El Hessen, T., Tuninetti, D., Belkhadir, A., and Banerjee, A. (2025, January 22\u201327). Channel Capacity Analysis with Nonlinear Effects of RF Power Amplifiers. Proceedings of the 2025 IEEE International Symposium on Information Theory (ISIT), Ann Arbor, MI, USA.","DOI":"10.1109\/ISIT63088.2025.11195697"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Stapmanns, J., Dias, C., Eilers, L., K\u00fchn, T., and Pfister, J.P. (2025). Phase Transitions of the Additive Uniform Noise Channel with Peak Amplitude and Cost Constraint. arXiv.","DOI":"10.1109\/TIT.2026.3687151"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"5469","DOI":"10.1109\/TIT.2025.3558841","article-title":"On the Best Approximation by Finite Gaussian Mixtures","volume":"71","author":"Ma","year":"2025","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1109\/TIT.1982.1056454","article-title":"Channel coding with multilevel\/phase signals","volume":"28","author":"Ungerboeck","year":"2003","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1426","DOI":"10.1109\/18.59937","article-title":"On the capacity of the Gaussian channel with a finite number of input levels","volume":"36","author":"Ozarow","year":"1990","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Dytso, A., Goldenbaum, M., Poor, H.V., and Shitz, S.S. (2017, January 25\u201330). A generalized Ozarow-Wyner capacity bound with applications. Proceedings of the 2017 IEEE International Symposium on Information Theory (ISIT), Aachen, Germany.","DOI":"10.1109\/ISIT.2017.8006690"},{"key":"ref_30","first-page":"246","article-title":"An information theoretical identity and a problem involving capacity","volume":"2","year":"1967","journal-title":"Stud. Sci. Math. Hung."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/5\/500\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:47:04Z","timestamp":1777448824000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/5\/500"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,28]]},"references-count":30,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2026,5]]}},"alternative-id":["e28050500"],"URL":"https:\/\/doi.org\/10.3390\/e28050500","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,28]]}}}