TL;DR
This paper investigates the size of specific sum-product sets involving a finite subset of real numbers, providing partial answers to a conjecture and establishing new bounds on related set cardinalities.
Contribution
It offers a partial resolution to A. Balog's conjecture and introduces new results on the cardinalities of sets like A:A+A, AA+AA, and A:A + A:A.
Findings
Partial answer to Balog's conjecture on AA+A.
New bounds on the sizes of A:A+A, AA+AA, and A:A + A:A.
Abstract
We give a partial answer to a conjecture of A. Balog, concerning the size of AA+A, where A is a finite subset of real numbers. Also, we prove several new results on the cardinality of A:A+A, AA+AA and A:A + A:A.
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