Please provide me some practical/real life applications for the sumsets of the sets of integers or real numbers. Please suggest me some useful references too..
Sumsets appear in many problems of additive combinatorics. The celebrated Lagrange's Four square theorem is nothing but a statement of sumsets, which says any natural number can be represented as the sum of four integer squares. You can write it as
4S=N where S=set of squares of integers and N= the set of naturals.
More generally, the waring's problems are the examples of sumsets.
Thank you Professor Akira Kanda and Professor Tapas Chatterjee for valuable remarks...
The sum is sometimes a direct sum, like in coordinate axes. R2 = (1,0) R + (0,1) R and this fact has a lot of applications (!) Another interesting fact is that R = Z + [0,1] and the decomposition is again unique. This leads to the functions integer part and fractional part. Also interesting, the sum Z + sqrt(2) Z is dense in R. These are the most easy (trivial) examples, but there are a lot of nice sumsets of reals!
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