It is well known that in K-user constant single-antenna interference channels K/2 degrees of freedom (DoF) can be achieved for almost all channel matrices. It is also known that almost all channel matrices admit K/2 DoF, but explicit conditions available guaranteeing K/2 DoF are satisfied only on a set of Lebesgue measure zero. We close this gap by identifying explicit conditions for K/2 DoF, which are satisfied for Lebesgue almost all channel matrices. We also provide a construction of corresponding asymptotically DoF-optimal input distributions. The main technical tool used is a recent breakthrough result by Hochman in fractal geometry. We conclude by discussing connections between interference alignment and additive combinatorics.
Helmut Bölcskei was born in Mödling, Austria on May 29, 1970, and received the Dipl.-Ing. and Dr. techn. degrees in electrical engineering from Vienna University of Technology, Vienna, Austria, in 1994 and 1997, respectively. In 1998 he was with Vienna University of Technology. From 1999 to 2001 he was a postdoctoral researcher in the Information Systems Laboratory, Department of Electrical Engineering, and in the Department of Statistics, Stanford University, Stanford, CA. He was in the founding team of Iospan Wireless Inc., a Silicon Valley-based startup company (acquired by Intel Corporation in 2002) specialized in multiple-input multiple-output (MIMO) wireless systems for high-speed Internet access, and was a co-founder of Celestrius AG, Zurich, Switzerland. From 2001 to 2002 he was an Assistant Professor of Electrical Engineering at the University of Illinois at Urbana-Champaign. He has been with ETH Zurich since 2002, where he is Professor of Electrical Engineering. He was a visiting researcher at Philips Research Laboratories Eindhoven, The Netherlands, ENST Paris, France, and the Heinrich Hertz Institute Berlin, Germany. His research interests are in information theory, mathematical signal processing, learning theory, and statistics.
He received the 2001 IEEE Signal Processing Society Young Author Best Paper Award, the 2006 IEEE Communications Society Leonard G. Abraham Best Paper Award, the 2010 Vodafone Innovations Award, the ETH “Golden Owl” Teaching Award, is a Fellow of the IEEE, a 2011 EURASIP Fellow, a 2013-2014 Distinguished Lecturer of the IEEE Information Theory Society, was an Erwin Schrödinger Fellow (1999-2001) of the Austrian National Science Foundation (FWF), and was included in the 2014 Thomson Reuters List of Highly Cited Researchers in Computer Science. He was a plenary speaker at several IEEE conferences and served as an associate editor of the IEEE Transactions on Information Theory, the IEEE Transactions on Signal Processing, the IEEE Transactions on Wireless Communications, and the EURASIP Journal on Applied Signal Processing. He was editor-in-chief of the IEEE Transactions on Information Theory during the period 2010-2013, and serves on the editorial boards of “Foundations and Trends in Networking”, “Foundations and Trends in Communications and Information Theory”, and the IEEE Signal Processing Magazine. He was TPC co-chair of the 2008 IEEE International Symposium on Information Theory and served on the Board of Governors of the IEEE Information Theory Society.