I would like to know some practical applications of additive combinatorics in the field of engineering and architecture. Please suggest some useful reference.
This is a good question with lots of possible answers.
In addition to the incisive answers already given, consider the excellent overview of additive combinatorics and its applications in
K. Bibak, Additive combinatorics with a view towards Computer Science and Cryptography, University of Waterloo, 2012:
http://cacr.uwaterloo.ca/techreports/2011/cacr2011-31.pdf
See, for example, Section 3, starting on page 10, on the sum-product problem theory and applications. There are 350 references to related books and papers, starting on page 16.
An excellent, in-depth study of additive combinatorics theory and some applications is given in
N.Ron-Zewi, Additive Combinatorics Methods in Computational Complexity, Ph.D. thesis, Technion, 2014:
http://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-get.cgi/2014/PHD/PHD-2014-10.pdf
See Section 7.1, starting on page 131, on combinatorial applications.
Even depends on what intersection of Additive Combinatorics you are interested in. Additive combinatorics crops up in numerous areas of Theoretical Computer Science for example (hence why a lot of combinatorists tend to work in CS).
A simple example is sphere-packing problems, and the theory of error correcting codes. Honestly, if there is something that needs to be counted or generated, combinatorics is a core part of an algorithm or parameters of an algorithm best suited for solving that problem. One superb example of this in the development of loopless algorithms. The algorithms of this type rely heavily on generating functions and being extremely careful on how to construct numeric sequences or their sizes play a vital role in the asymptotics of the algorithm. Usually all of these come back to a clever way of counting constructively a set (and combinatorial arguments).
I don't have any "readings" really for this, but some things here may help. Most books on Combinatorial Algorithm will set aside some parts of it to additive combinatorics. All of these things do come up all the time if one wants to solve these hard mathematical problems via computation.
One interesting use arose a while back: more than 20 years ago: some electrical engineers were trying to tag data with integers, so that when they performed arithmetic (addition, multiplication, subtraction) on the data, and performed the same operations on the tags, the results on the tags could be checked to see that the correct operations on the data had been performed.
They wanted a set of integers in {0, 2, ..., n-1} for which all pairwise sums, products and differences were distinct. I showed them an easy proof that just getting distinct sums would limit them to a set of size about sqrt{n}, and gave them a fairly easy modification of the standard construction of difference sets (mod p) that constructed a set of size about sqrt{n}/log(n). Unfortunately, the size of set they wanted was too big to fit into 32bit integers, and they abandoned the project. I still found it a really fascinating application of combinatorial number theory (not necessarily additive combinatorics,, but...)
This is a good question with lots of possible answers.
In addition to the incisive answers already given, consider the excellent overview of additive combinatorics and its applications in
K. Bibak, Additive combinatorics with a view towards Computer Science and Cryptography, University of Waterloo, 2012:
http://cacr.uwaterloo.ca/techreports/2011/cacr2011-31.pdf
See, for example, Section 3, starting on page 10, on the sum-product problem theory and applications. There are 350 references to related books and papers, starting on page 16.
An excellent, in-depth study of additive combinatorics theory and some applications is given in
N.Ron-Zewi, Additive Combinatorics Methods in Computational Complexity, Ph.D. thesis, Technion, 2014:
http://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-get.cgi/2014/PHD/PHD-2014-10.pdf
See Section 7.1, starting on page 131, on combinatorial applications.
Thank you very much Respected Professors, for your valuable remarks and inputs..
Dear @Sudev, some engineering applications are listed in this free e-books, link follows.
The application of additive combinatorics is given by the next two case studies, in the area of Computer Science and Engineering, as well as in Metabolic Engineering!
http://www.3000ebooks.org/download/258592-additive-combinatorics.html
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3962077/
http://cseweb.ucsd.edu/~slovett/files/addcomb-survey.pdf
To enable agility by additive manufacturing (complexity, agility, efficiency) implementing concurrent, hybrid processes, considering the design space with topology organization,
production constraints, and optimization of variables and materials. Innovation is better through additive manufacturing, such as using Additive Topology Optimized Manufacturing (ATOM) which helped place brackets on the Black Hawk helicopter, and drastically reduces wasted material.
http://www.mse.gatech.edu/sites/default/files/2013%20Industry%20Symposium%20Proceedings.pdf
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