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New Sum-Product Estimates for Real and Complex Numbers
A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite set A of positive real numbers, it is true that |{a+bc+d:a,b,c,d∈A}|≥2|A|2-1.As a consequence of this result, it is also established that |4k-1A(k)|:=|A…A⏟ktimes+⋯+A…A⏟4k-1times|≥|A|k.Later on, it is shown that both of these bounds hold in the case when A is a finite set of complex numbers, although with smaller multiplicative constants. © 2015, Springer Science+Business Media New York
A sum-product theorem in function fields
Let be a finite subset of \ffield, the field of Laurent series in
over a finite field . We show that for any there
exists a constant dependent only on and such that
. In particular such a result is
obtained for the rational function field . Identical results
are also obtained for finite subsets of the -adic field for
any prime .Comment: Simplification of argument and note that methods also work for the
p-adic
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