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Diameter of sets and measure of sumsets. (English) Zbl 0737.28006

We define the diameter of a bounded set \(B\) as \(D(B)=\sup B-\inf B\). Let \(A\), \(B\) be bounded sets of reals with \(\underline\mu(A)=a\), \(\underline\mu(B)=b\), \(D(B)=D\) (\(\underline\mu\) denotes inner Lebesgue measure). We establish the inequality \[ \underline\mu(A+B)\geq\min (a+D,(\sqrt a+\sqrt{b/2})^ 2). \] In particular, if \(a\geq D^ 2/(2b)\), then \[ \underline\mu(A+B)\geq a+D.\tag{1} \] This improves the familiar inequality \(\underline\mu(A+B)\geq\underline\mu(A)+\underline\mu(B)\) for \(a\) not too small. A more complicated bound is also given that yields the best possible lower estimate of \(\underline\mu(A+B)\) in terms of \(a\), \(b\) and \(D\), and also gives the exact bound after which (1) holds. If \(A\) is a long interval, then there is equality in (1).
Reviewer: I.Z.Ruzsa

MSC:

28A75 Length, area, volume, other geometric measure theory
26B15 Integration of real functions of several variables: length, area, volume
11P99 Additive number theory; partitions
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References:

[1] Kemperman, J. H. B.: On products of sets in a locally compact group. Fundamenta Math.56, 51-68 (1964). · Zbl 0125.28901
[2] Raikov, D. A.: On the addition of point sets in the sense of Schnirelmann (in Russian). Mat. Sbornik5, 425-440 (1939). · Zbl 0022.21003
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