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topics

Self-similar sets and measures and Bernoulli convolutions

  1. Amir Algom, Simon Baker and Pablo Shmerkin. On normal numbers and self-similar measures. Adv. Math. 399 (2022), Paper No. 108276. https://arxiv.org/abs/2111.10082 .

  2. Aleksi Pyörälä, Pablo Shmerkin, Ville Suomala and Meng Wu. Covering the Sierpinski carpet with tubes. Preprint. https://arxiv.org/abs/2006.00499.

  3. Pablo Shmerkin. On Furstenberg's intersection conjecture, self-similar measures, and the L^q norms of convolutions. Ann. of Math. (2) 189 (2019), no. 2, 319--391. https://arxiv.org/abs/1609.07802 .

  4. Santiago Saglietti, Pablo Shmerkin and Boris Solomyak. Absolute continuity of non-homogeneous self-similar measures. Adv. Math. 335 (2018), 60--110. https://arxiv.org/abs/1709.05092 .

  5. Carolina Mosquera and Pablo Shmerkin. Self-similar measures: asymptotic bounds for the dimension and Fourier decay or smooth images. Ann. Acad. Sci. Fenn. Math. 43 (2018), no. 2, 823--834. https://arxiv.org/abs/1710.06812

  6. Pablo Shmerkin and Boris Solomyak. Absolute continuity of complex Bernoulli convolutions. Math. Proc. Cambridge Philos. Soc. 161 (2016), no. 3, 435--453.  http://arxiv.org/abs/1504.00631 .

  7. Pablo Shmerkin. Projections of Self-Similar and Related Fractals: A Survey of Recent Developments. In Fractal Geometry and Stochastics V. Progress in Probability, Vol. 70. Birkhäuser Basel. http://arxiv.org/abs/1501.00875 .

  8. Pablo Shmerkin and Boris Solomyak.  Absolute continuity of self-similar measures, their projections and convolutions. Trans. Amer. Math. Soc. 368(2016), no. 7, 5125--5151. http://arxiv.org/abs/1406.0204 .

  9. Pablo Shmerkin. On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions. Geom. Funct. Anal. 24 (2014), no. 3, 946--958. http://arxiv.org/abs/1303.3992 .

  10. Michael Hochman and Pablo Shmerkin. Local entropy averages and projections of fractal measures. Ann. of Math. (2) 175 (2012), no. 3, 1001--1059. http://arxiv.org/abs/0910.1956 .

  11. Fedor Nazarov, Yuval Peres and Pablo Shmerkin. Convolutions of Cantor measures without resonance. Israel J. Math. 187 (2012), 93--116. http://arxiv.org/abs/0905.3850 .

  12. Thomas Jordan, Pablo Shmerkin and Boris Solomyak. Multifractal structure of Bernoulli convolutions. Math. Proc. Cambridge Philos. Soc. 151 (2011), no. 3, 521--539. http://arxiv.org/abs/1011.1938 .

  13. Yuval Peres and Pablo Shmerkin. Resonance between Cantor sets. Ergodic Theory Dynam. Systems 29 (2009), no. 1, 201--221. http://arxiv.org/abs/0705.2628 .

  14. Pablo Shmerkin. A modified multifractal formalism for a class of self-similar measures with overlap. Asian J. Math. 9 (2005), no. 3, 323--348. http://arxiv.org/abs/math/0408047 .

selfsimilar

Self-affine sets and the subadditive thermodynamic formalism

  1. Ian D. Morris and Pablo Shmerkin. On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems. Trans. Amer. Math. Soc. 371 (2019), no. 3, 1547--1582. https://arxiv.org/abs/1602.08789 .

  2. Jonathan Fraser and Pablo Shmerkin.  On the dimensions of a family of overlapping self-affine carpets. Ergodic Theory Dynam. Systems 36 (2016), no. 8, 2463--2481. http://arxiv.org/abs/1405.4919 .

  3. De-Jun Feng and Pablo Shmerkin. Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles. Geom. Funct. Anal. 24 (2014), no. 4, 1101--1128. http://arxiv.org/abs/1311.4241.

  4. Pablo Shmerkin . Self-affine sets and the continuity of subadditive pressure. In Geometry and Analysis of fractals.  Springer Proceedings in Mathematics & Statistics, Vol. 88.  http://arxiv.org/abs/1309.4730 .

  5. Andrew Ferguson, Thomas Jordan, and Pablo Shmerkin. The Hausdorff dimension of the projections of self-affine carpets. Fund. Math. 209 (2010), no. 3, 193--213. http://arxiv.org/abs/0903.2216 .

  6. Antti Käenmäki and Pablo Shmerkin. Overlapping self-affine sets of Kakeya type. Ergodic Theory Dynam. Systems 29 (2009), no. 3, 941--965. http://arxiv.org/abs/0710.0442 .

  7. Pablo Shmerkin and Boris Solomyak. Zeros of {-1,0,1} power series and connectedness loci for self-affine sets. Experiment. Math. 15 (2006), no. 4, 499--511. http://arxiv.org/abs/math/0504545 .

  8. Pablo Shmerkin. Overlapping self-affine sets. Indiana Univ. Math. J. 55 (2006), no. 4, 1291--1331. http://arxiv.org/abs/math/0408203 .

selfaffine

Distance sets and geometric configurations

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  1. Tuomas Orponen, Pablo Shmerkin and Hong Wang. Kaufman and Falconer estimates for radial projections and a continuum version of Beck's Theorem. https://arxiv.org/abs/2209.00348.

  2. Pablo Shmerkin and Hong Wang. On the distance sets spanned by sets of dimension d/2 in R^d. https://arxiv.org/abs/2112.09044 .

  3. Pablo Shmerkin. A nonlinear version of Bourgain's projection theorem. To appear in J. Eur. Math. Soc. (JEMS). https://arxiv.org/abs/2003.01636 .

  4. Pablo Shmerkin and Han Yu. On sets containing a unit distance in every direction. Discrete Analysis, 2021:5, 13p.. https://arxiv.org/abs/1912.01523.

  5. Pablo Shmerkin. Improved bounds for the dimensions of planar distance sets. J. Fractal Geom. 8 (2021), no. 1, 27--51.  https://arxiv.org/abs/1811.03379 .

  6. Tamás Keleti and Pablo Shmerkin. New bounds on the dimensions of planar distance sets.  Geom. Funct. Anal. 29 (2019), no. 6, 1886--1948. https://arxiv.org/abs/1801.08745 .

  7. Andrea Olivo and Pablo Shmerkin. Maximal operators for cube skeletons. Ann. Acad. Sci. Fenn. Math. 45 (2020), no. 1, 467--478. https://arxiv.org/abs/1807.05280.

  8. Pablo Shmerkin. On the Hausdorff dimension of pinned distance sets. Israel J. Math. 230 (2019), no. 2, 949--972. https://arxiv.org/abs/1706.00131 .

  9. Pablo Shmerkin and Ville Suomala.  Spatially independent martingales, intersections, and applications. Mem. Amer. Math. Soc. 251 (2018), no. 1195, v+102 pp. http://arxiv.org/abs/1409.6707 .

  10. Tamás Keleti, Dániel Nagy and  Pablo Shmerkin. Squares and their centers. J. Anal. Math. 134 (2018), no. 2, 643-669.  http://arxiv.org/abs/1408.1029 .

  11. Pablo Shmerkin. On distance sets, box-counting and Ahlfors-regular sets. Discrete Analysis, 2017:9, 22p. http://discreteanalysisjournal.com/article/1643-on-distance-sets-box-counting-and-ahlfors-regular-se...

distancesets
GMT

Geometric measure theory

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  1. Tuomas Orponen and Pablo Shmerkin. Projections, Furstenberg sets, and the ABC sum-product problem. https://arxiv.org/abs/2301.10199 .

  2. Pablo Shmerkin and Hong Wang. Dimensions of Furstenberg sets and an extension of Bourgain's projection theorem. https://arxiv.org/abs/2211.13363 .

  3. Tuomas Orponen, Pablo Shmerkin and Hong Wang. Kaufman and Falconer estimates for radial projections and a continuum version of Beck's Theorem. https://arxiv.org/abs/2209.00348.

  4. Tuomas Orponen and Pablo Shmerkin. On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane. To appear in Duke Math. J. https://arxiv.org/abs/2106.03338.

  5. Aleksi Pyörälä, Pablo Shmerkin, Ville Suomala and Meng Wu. Covering the Sierpinski carpet with tubes. Preprint. https://arxiv.org/abs/2006.00499.

  6. Pablo Shmerkin. A nonlinear version of Bourgain's projection theorem. To appear in J. Eur. Math. Soc. (JEMS). https://arxiv.org/abs/2003.01636 .

  7. Pablo Shmerkin and Han Yu. On sets containing a unit distance in every direction. Discrete Analysis, 2021:5, 13p.. https://arxiv.org/abs/1912.01523.

  8. Kornélia Héra, Pablo Shmerkin and Alexia Yavicoli. An improved bound for the dimension of (α,2α)-Furstenberg sets. Rev. Mat. Iberoam. 38 (2022), no. 1, 295–322. https://arxiv.org/abs/2001.11304 .

  9. Kenneth J. Falconer, Jonathan M. Fraser and Pablo Shmerkin. Assouad dimension influences the box and packing dimensions of orthogonal projections. J. Fractal Geom., accepted for publication. https://arxiv.org/abs/1911.04857.

  10. Andrea Olivo and Pablo Shmerkin. Maximal operators for cube skeletons. Ann. Acad. Sci. Fenn. Math. 45 (2020), no. 1, 467--478. https://arxiv.org/abs/1807.05280.

  11. Eino Rossi and Pablo Shmerkin. On measures that improve $L^q$ dimension under convolution. Rev. Mat. Iberoam., accepted for publication, 2019. https://arxiv.org/abs/1812.05660.

  12. Eino Rossi and Pablo Shmerkin. Hölder coverings of sets of small dimension.  J. Fractal Geom. 6 (2019), no. 3, 285--299. https://arxiv.org/abs/1702.01130 .

  13. Pablo Shmerkin and Ville Suomala.  Spatially independent martingales, intersections, and applications. Mem. Amer. Math. Soc. 251 (2018), no. 1195, v+102 pp. http://arxiv.org/abs/1409.6707 .

  14. Tamás Keleti, Dániel Nagy and  Pablo Shmerkin. Squares and their centers. J. Anal. Math. 134 (2018), no. 2, 643-669.  http://arxiv.org/abs/1408.1029 .

  15. Pablo Shmerkin and Ville Suomala. Sets which are not tube null and intersection properties of random measures. J. Lond. Math. Soc. (2) 91 (2015), no. 2, 405--422.  http://arxiv.org/abs/1204.5883v2.

  16. Antti Käenmäki,  Tuomas Sahlsten and Pablo Shmerkin. Dynamics of the scenery flow and geometry of measures. Proc. Lond. Math. Soc. (3) 110 (2015), no. 5, 1248--1280. http://arxiv.org/abs/1401.0231.

  17. Antti Käenmäki,  Tuomas Sahlsten and Pablo Shmerkin. Structure of distributions generated by the scenery flow.  J. Lond. Math. Soc. (2) 91 (2015), no. 2, 464--494. http://arxiv.org/abs/1312.2567 .

  18. Ignacio Garcia and Pablo Shmerkin. On packing measures and a theorem of Besicovitch. Proc. Amer. Math. Soc. 142 (2014), no. 8, 2661--2669. http://arxiv.org/abs/1205.6224 .

  19. Tuomas Sahlsten, Pablo Shmerkin and Ville Suomala. Dimension, entropy and the local distribution of measures. J. Lond. Math. Soc. (2) 87 (2013), no. 1, 247--268. http://arxiv.org/abs/1110.6011 .

  20. Pablo Shmerkin. The dimension of weakly mean porous measures: a probabilistic approach. Int. Math. Res. Not. IMRN 2012, no. 9, 2010--2033. http://arxiv.org/abs/1010.1394 .

  21. Pablo Shmerkin. Porosity, dimension, and local entropies: a survey. Rev. Un. Mat. Argentina 52 (2011), no. 2, 81--103. http://arxiv.org/abs/1110.5682 .

  22. Jörg Schmeling and Pablo Shmerkin. On the dimension of iterated sumsets. Recent developments in fractals and related fields, 55--72, Appl. Numer. Harmon. Anal.,Birkhäuser Boston, Inc., Boston, MA, 2010. http://arxiv.org/abs/0906.1537 .

ergodictheory

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  1. Amir Algom, Simon Baker and Pablo Shmerkin. On normal numbers and self-similar measures. Adv. Math. 399 (2022), Paper No. 108276. https://arxiv.org/abs/2111.10082 .

  2. Pablo Shmerkin. On Furstenberg's intersection conjecture, self-similar measures, and the L^q norms of convolutions. Ann. of Math. (2) 189 (2019), no. 2, 319--391. https://arxiv.org/abs/1609.07802 .

  3. Daniel Galicer, Santiago Saglietti, Pablo Shmerkin and Alexia Yavicoli. L^q dimensions and projections of random measures. Nonlinearity 29 (2016), no. 9, 2609--2640. https://arxiv.org/abs/1504.04893 .

  4. Michael Hochman and Pablo Shmerkin. Equidistribution from fractal measures. Invent. Math. 202 (2015), no. 1, 427--479 . http://arxiv.org/abs/1302.5792

  5. Antti Käenmäki,  Tuomas Sahlsten and Pablo Shmerkin. Dynamics of the scenery flow and geometry of measures. Proc. Lond. Math. Soc. (3) 110 (2015), no. 5, 1248--1280. http://arxiv.org/abs/1401.0231.

  6. Antti Käenmäki,  Tuomas Sahlsten and Pablo Shmerkin. Structure of distributions generated by the scenery flow.  J. Lond. Math. Soc. (2) 91 (2015), no. 2, 464--494. http://arxiv.org/abs/1312.2567 .

  7. De-Jun Feng and Pablo Shmerkin. Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles. Geom. Funct. Anal. 24 (2014), no. 4, 1101--1128. http://arxiv.org/abs/1311.4241.

  8. Pablo Shmerkin . Self-affine sets and the continuity of subadditive pressure. In Geometry and Analysis of fractals.  Springer Proceedings in Mathematics & Statistics, Vol. 88.  http://arxiv.org/abs/1309.4730 .

  9. Michael Hochman and Pablo Shmerkin. Local entropy averages and projections of fractal measures. Ann. of Math. (2) 175 (2012), no. 3, 1001--1059. http://arxiv.org/abs/0910.1956 .

  10. Fedor Nazarov, Yuval Peres and Pablo Shmerkin. Convolutions of Cantor measures without resonance. Israel J. Math. 187 (2012), 93--116. http://arxiv.org/abs/0905.3850 .

  11. Yuval Peres and Pablo Shmerkin. Resonance between Cantor sets. Ergodic Theory Dynam. Systems 29 (2009), no. 1, 201--221. http://arxiv.org/abs/0705.2628 .

additivecombinatorics

Additive combinatorics and fractal geometry

  1. Tuomas Orponen, Nicolas de Saxcé and Pablo Shmerkin. On the Fourier decay of multiplicative convolutions. https://arxiv.org/abs/2309.03068 .

  2. Tuomas Orponen and Pablo Shmerkin. Projections, Furstenberg sets, and the ABC sum-product problem. https://arxiv.org/abs/2301.10199 .

  3. Pablo Shmerkin and Hong Wang. Dimensions of Furstenberg sets and an extension of Bourgain's projection theorem. https://arxiv.org/abs/2211.13363 .

  4. Tuomas Orponen and Pablo Shmerkin. On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane. To appear in Duke Math. J., https://arxiv.org/abs/2106.03338.

  5. Pablo Shmerkin. A nonlinear version of Bourgain's projection theorem. To appear in J. Eur. Math. Soc. (JEMS). https://arxiv.org/abs/2003.01636 .

  6. Jonathan M. Fraser, Pablo Shmerkin and Alexia Yavicoli. Improved bounds on the dimensions of sets that avoid approximate arithmetic progressions. J. Fourier Anal. Appl. 27 (2021), no. 1, Paper No. 4, 14 pp. https://arxiv.org/abs/1910.10074 .

  7. Pablo Shmerkin. On the packing dimension of Furstenberg sets. J. Anal. Math., 146 (2022), no. 1, 351–364 . https://arxiv.org/abs/2006.15569 .

  8. Pablo Shmerkin and Ville Suomala. Patterns in random fractals. Amer. J. Math. 142 (2020), no. 3, 683--749. https://arxiv.org/abs/1703.09553 .

  9. Pablo Shmerkin. On Furstenberg's intersection conjecture, self-similar measures, and the L^q norms of convolutions. Ann. of Math. (2) 189 (2019), no. 2, 319--391. https://arxiv.org/abs/1609.07802 .

  10. Pablo Shmerkin. Salem Sets with No Arithmetic Progressions. Int. Math. Res. Not. IMRN 2017, no. 7, 1929--1941. http://arxiv.org/abs/1510.07596 .

  11. Michael Hochman and Pablo Shmerkin. Local entropy averages and projections of fractal measures. Ann. of Math. (2) 175 (2012), no. 3, 1001--1059. http://arxiv.org/abs/0910.1956 .

  12. Jörg Schmeling and Pablo Shmerkin. On the dimension of iterated sumsets. Recent developments in fractals and related fields, 55--72, Appl. Numer. Harmon. Anal.,Birkhäuser Boston, Inc., Boston, MA, 2010. http://arxiv.org/abs/0906.1537 .

Harmonic Analysis

  1. Tuomas Orponen, Nicolas de Saxcé and Pablo Shmerkin. On the Fourier decay of multiplicative convolutions. https://arxiv.org/abs/2309.03068 .

  2. Pablo Shmerkin and Ville Suomala. New bounds for Cantor maximal operators. Rev. Un. Mat. Argentina 64 (2022), no 1, 69--86. https://arxiv.org/abs/2106.14818.

  3. Eino Rossi and Pablo Shmerkin. On measures that improve $L^q$ dimension under convolution. Rev. Mat. Iberoam. 36 (2020), no. 7, 2217--2236.

  4. Andrea Olivo and Pablo Shmerkin. Maximal operators for cube skeletons. Ann. Acad. Sci. Fenn. Math. 45 (2020), no. 1, 467--478. https://arxiv.org/abs/1807.05280.

  5. Pablo Shmerkin and Ville Suomala.  Spatially independent martingales, intersections, and applications. Mem. Amer. Math. Soc. 251 (2018), no. 1195, v+102 pp. http://arxiv.org/abs/1409.6707 .

  6. Carolina Mosquera and Pablo Shmerkin. Self-similar measures: asymptotic bounds for the dimension and Fourier decay or smooth images. Ann. Acad. Sci. Fenn. Math. 43 (2018), no. 2, 823--834. https://arxiv.org/abs/1710.06812

  7. Pablo Shmerkin and Ville Suomala. Sets which are not tube null and intersection properties of random measures. J. Lond. Math. Soc. (2) 91 (2015), no. 2, 405--422.  http://arxiv.org/abs/1204.5883v2.

harmonic-analysis

Random fractals and applications

  1. Pablo Shmerkin and Ville Suomala. New bounds for Cantor maximal operators. Rev. Un. Mat. Argentina 64 (2022), no 1, 69--86. https://arxiv.org/abs/2106.14818 .

  2. Pablo Shmerkin and Ville Suomala. Patterns in random fractals. Amer. J. Math. 142 (2020), no. 3, 683--749. https://arxiv.org/abs/1703.09553 .

  3. Pablo Shmerkin and Ville Suomala.  Spatially independent martingales, intersections, and applications. Mem. Amer. Math. Soc. 251 (2018), no. 1195, v+102 pp. http://arxiv.org/abs/1409.6707 .

  4. Pablo Shmerkin. Salem Sets with No Arithmetic Progressions. Int. Math. Res. Not. IMRN 2017, no. 7, 1929--1941. http://arxiv.org/abs/1510.07596 .

  5. Daniel Galicer, Santiago Saglietti, Pablo Shmerkin and Alexia Yavicoli. L^q dimensions and projections of random measures. Nonlinearity 29 (2016), no. 9, 2609--2640. https://arxiv.org/abs/1504.04893 .

  6. Pablo Shmerkin and Ville Suomala. Sets which are not tube null and intersection properties of random measures. J. Lond. Math. Soc. (2) 91 (2015), no. 2, 405--422.  http://arxiv.org/abs/1204.5883v2.

  7. Ida Arhosalo, Esa Järvenpää, Maarit Järvenpää, Michał Rams and Pablo Shmerkin . Visible parts of fractal percolation. Proc. Edinb. Math. Soc. (2) 55 (2012), no. 2, 311--331. http://arxiv.org/abs/0911.3931 .

randomfractals
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