A combinatorial proof of a sumset conjecture of Furstenberg
We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if log r / log s is irrational and X and Y are × r - and × s -invariant subsets of [0, 1], respectively, then dim H ( X + Y ) = min ( 1 , dim H X + dim H Y ) . Our main result yields info...
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Veröffentlicht in: | Combinatorica (Budapest. 1981) 2023-04, Vol.43 (2), p.299-328 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if
log
r
/
log
s
is irrational and
X
and
Y
are
×
r
- and
×
s
-invariant subsets of [0, 1], respectively, then
dim
H
(
X
+
Y
)
=
min
(
1
,
dim
H
X
+
dim
H
Y
)
. Our main result yields information on the size of the sumset
λ
X
+
η
Y
uniformly across a compact set of parameters at fixed scales. The proof is combinatorial and avoids the machinery of local entropy averages and CP-processes, relying instead on a quantitative, discrete Marstrand projection theorem and a subtree regularity theorem that may be of independent interest. |
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ISSN: | 0209-9683 1439-6912 |
DOI: | 10.1007/s00493-023-00008-9 |