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We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than 1, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In particular, we prove that if dimH (A) > 1, then the set of distances spanned by points of A has Hausdorff dimension at least 40/57 > 0.7 and there are many y ∈ A such that the pinned distance set {\x - y\: x ∈ A} has Hausdorff dimension at least 29/42 and lower box-counting dimension at least 40/57. We use the approach and many results from the earlier work of Keleti and Shmerkin, but incorporate estimates from the recent work of Guth, Iosevich, Ou and Wang as additional input. [ABSTRACT FROM AUTHOR]

The Lq dimensions, for 1 < q < 8, quantify the degree of smoothness of a measure. We study the following problem on the real line: when does the Lq dimension improve under convolution? This can be seen as a variant of the well-known Lp-improving property. Our main result asserts that uniformly perfect measures (which include Ahlfors-regular measures as a proper subset) have the property that convolving with them results in a strict increase of the Lq dimension. We also study the case q = 8, which corresponds to the supremum of the Frostman exponents of the measure. We obtain consequences for repeated convolutions and for the box dimension of sumsets. Our results are derived from an inverse theorem for the Lq norms of convolutions due to the second author. [ABSTRACT FROM AUTHOR]

We construct compact Salem sets in R/Z of any dimension (including 1), which do not contain any arithmetic progressions of length 3. Moreover, the sets can be taken to be Ahlfors regular if the dimension is less than 1, and the measure witnessing the Fourier decay can be taken to be Frostman in the case of dimension 1. This is in sharp contrast to the situation in the discrete setting (where Fourier uniformity is well known to imply existence of progressions) and helps clarify a result of Łaba and Pramanik on pseudorandom subsets of R which do contain progressions. [ABSTRACT FROM AUTHOR]

We consider the dimensions of a family of self-affine sets related to the Bedford–McMullen carpets. In particular, we fix a Bedford–McMullen system and then randomize the translation vectors with the stipulation that the column structure is preserved. As such, we maintain one of the key features in the Bedford–McMullen set-up in that alignment causes the dimensions to drop from the affinity dimension. We compute the Hausdorff, packing and box dimensions outside of a small set of exceptional translations, and also for some explicit translations even in the presence of overlapping. Our results rely on, and can be seen as a partial extension of, Hochman’s recent work on the dimensions of self-similar sets and measures. [ABSTRACT FROM PUBLISHER]

We present a general approach to the study of the local distribution of measures on Euclidean spaces, based on local entropy averages. As concrete applications, we unify, generalize and simplify a number of recent results on local homogeneity, porosity and conical densities of measures. [ABSTRACT FROM PUBLISHER]

Using probabilistic ideas, we prove that if μ is a mean porous measure on , then the packing dimension of μ is strictly smaller than n. Moreover, we give an explicit bound for the packing dimension, which is asymptotically sharp in the case of small porosity. This result was stated in [D. B. Beliaev and S. K. Smirnov, “On dimension of porous measures”, Mathematische Annalen 323 (2002): 123–41], but the proof given there is not correct. We also give estimates on the dimension of weakly mean porous measures, which improve another result of Beliaev and Smirnov. [ABSTRACT FROM PUBLISHER]

Let { f i (x) = s i ⋅ x + t i } be a self-similar IFS on R and let β > 1 be a Pisot number. We prove that if log ⁡ | s i | log ⁡ β ∉ Q for some i then for every C 1 diffeomorphism g and every non-atomic self-similar measure μ , the measure gμ is supported on numbers that are normal in base β. [ABSTRACT FROM AUTHOR]