TY - JOUR AU - Balka, Richárd AU - Elekes, Márton AU - Kiss, Viktor AU - Nagy, Donát AU - Poór, Márk TI - Compact sets with large projections and nowhere dense sumset JF - NONLINEARITY J2 - NONLINEARITY VL - 36 PY - 2023 IS - 10 SP - 5190 EP - 5215 PG - 26 SN - 0951-7715 DO - 10.1088/1361-6544/acebae UR - https://m2.mtmt.hu/api/publication/34126755 ID - 34126755 AB - We answer a question of Banakh, Jabłońska and Jabłoński by showing that for d ⩾ 2 there exists a compact set K ⊆ R d such that the projection of K onto each hyperplane is of non-empty interior, but K + K is nowhere dense. The proof relies on a random construction. A natural approach in the proofs is to construct such a K in the unit cube with full projections, that is, such that the projections of K agree with that of the unit cube. We investigate the generalization of these problems for projections onto various dimensional subspaces as well as for ℓ -fold sumsets. We obtain numerous positive and negative results, but also leave open many interesting cases. We also show that in most cases if we have a specific example of such a compact set then actually the generic (in the sense of Baire category) compact set in a suitably chosen space is also an example. Finally, utilizing a computer-aided construction, we show that the compact set in the plane with full projections and nowhere dense sumset can be self-similar. LA - English DB - MTMT ER - TY - JOUR AU - Homa, Gábor AU - Balka, Richárd AU - Bernád, J.Z. AU - Károly, M. AU - Csordás, András TI - Newton’s identities and positivity of trace class integral operators JF - JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL J2 - J PHYS A-MATH THEOR VL - 56 PY - 2023 IS - 14 PG - 14 SN - 1751-8113 DO - 10.1088/1751-8121/acc147 UR - https://m2.mtmt.hu/api/publication/33720054 ID - 33720054 N1 - Export Date: 08 March 2024 LA - English DB - MTMT ER - TY - JOUR AU - Balka, Richárd AU - Elekes, Márton AU - Kiss, Viktor TI - Stability and measurability of the modified lower dimension JF - PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY J2 - P AM MATH SOC VL - 150 PY - 2022 IS - 9 SP - 3889 EP - 3898 PG - 10 SN - 0002-9939 DO - 10.1090/proc/16029 UR - https://m2.mtmt.hu/api/publication/32734627 ID - 32734627 N1 - Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15, Budapest, H-1053, Hungary Eötvös Loránd University, Institute of Mathematics, Pázmány Péter s. 1/c, Budapest, 1117, Hungary Export Date: 19 January 2023 Funding details: Magyar Tudományos Akadémia, MTA Funding details: Mount Allison University, MTA Funding details: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, NKFIH, 124749, 128273, 129211 Funding text 1: Received by the editors November 14, 2021. 2020 Mathematics Subject Classification. Primary 28A75, 28A20. Key words and phrases. Modified lower dimension, finite stability, measurability, Baire class. The first author was supported by the MTA Premium Postdoctoral Research Program and the National Research, Development and Innovation Office – NKFIH, grant no. 124749. The second author was supported by the National Research, Development and Innovation Office – NKFIH, grants no. 124749 and 129211. The third author was supported by the National Research, Development and Innovation Office – NKFIH, grants no. 124749, 129211, and 128273. Funding text 2: The first author was supported by the MTA Premium Postdoctoral Research Program and the National Research, Development and Innovation Office - NKFIH, grant no. 124749. The second author was supported by the National Research, Development and Innovation Office - NKFIH, grants no. 124749 and 129211. The third author was supported by the National Research, Development and Innovation Office - NKFIH, grants no. 124749, 129211, and 128273. AB - The lower dimension dimL is the dual concept of the Assouad dimension. As it fails to be monotonic, Fraser and Yu introduced the modified lower dimension dimML by making the lower dimension monotonic with the simple formula dimMLX=sup{dimLE:E⊂X}. As our first result we prove that the modified lower dimension is finitely stable in any metric space, answering a question of Fraser and Yu. We prove a new, simple characterization for the modified lower dimension. For a metric space X let K(X) denote the metric space of the non-empty compact subsets of X endowed with the Hausdorff metric. As an application of our characterization, we show that the map dimML:K(X)→[0,∞] is Borel measurable. More precisely, it is of Baire class 2, but in general not of Baire class 1. This answers another question of Fraser and Yu. Finally, we prove that the modified lower dimension is not Borel measurable defined on the closed sets of ℓ1 endowed with the Effros Borel structure. LA - English DB - MTMT ER - TY - JOUR AU - Balka, Richárd AU - Elekes, Márton AU - Kiss, Viktor AU - Poór, Márk TI - Singularity of maps of several variables and a problem of Mycielski concerning prevalent homeomorphisms JF - ADVANCES IN MATHEMATICS J2 - ADV MATH VL - 385 PY - 2021 SN - 0001-8708 DO - 10.1016/j.aim.2021.107773 UR - https://m2.mtmt.hu/api/publication/32012587 ID - 32012587 N1 - Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13–15, Budapest, H-1053, Hungary Eötvös Loránd University, Institute of Mathematics, Pázmány Péter s. 1/c, Budapest, 1117, Hungary Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, Jerusalem, 91904, Israel Export Date: 18 October 2022 Correspondence Address: Balka, R.; Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13–15, Hungary; email: balka.richard@renyi.hu Funding details: Magyar Tudományos Akadémia, MTA, 128273 Funding details: Emberi Eroforrások Minisztériuma, EMMI, ELTE/14325/272 Funding details: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, NKFIH, 113047, 124749, 129211 Funding details: National Research, Development and Innovation Office Funding text 1: The authors were supported by the National Research, Development and Innovation Office – NKFIH, grants no. 113047 , 129211 and 124749 . The first author was supported by the MTA Premium Postdoctoral Research Program grant no. PREMIUM–2018–302 . The third author was supported by the National Research, Development and Innovation Office – NKFIH, grant no. 128273 . The fourth author was supported through the New National Excellence Program of the Ministry of Human Capacities grant no. ELTE/14325/272 (2019) . LA - English DB - MTMT ER - TY - JOUR AU - Omer, Angel AU - Balka, Richárd AU - Máthé, András AU - Yuval, Peres TI - Restrictions of Hölder continuous functions JF - TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY J2 - T AM MATH SOC VL - 370 PY - 2018 IS - 6 SP - 4223 EP - 4247 PG - 25 SN - 0002-9947 DO - 10.1090/tran/7126 UR - https://m2.mtmt.hu/api/publication/3340900 ID - 3340900 N1 - Export Date: 10 November 2020 Funding details: Natural Sciences and Engineering Research Council of Canada, NSERC Funding details: 104178 Funding details: Leverhulme Trust Funding text 1: Received by the editors April 19, 2015 and, in revised form, November 11, 2016. 2010 Mathematics Subject Classification. Primary 26A16, 26A45, 28A78, 54E52, 60G17, 60G22, 60J65. Key words and phrases. Fractional Brownian motion, Hölder continuous, restriction, bounded variation, Hausdorff dimension, box dimension, Minkowski dimension, self-affine function, generic, typical, Baire category. The first author was supported in part by NSERC. The second and third authors were supported by the National Research, Development and Innovation Office-NKFIH, 104178. The third author was also supported by the Leverhulme Trust. LA - English DB - MTMT ER - TY - JOUR AU - Balka, Richárd AU - Tómács, Tibor TI - Baum-Katz type theorems with exact threshold JF - STOCHASTICS J2 - STOCHASTICS VL - 90 PY - 2018 IS - 4 SP - 473 EP - 503 PG - 31 SN - 1744-2508 DO - 10.1080/17442508.2017.1366490 UR - https://m2.mtmt.hu/api/publication/3260290 ID - 3260290 LA - English DB - MTMT ER - TY - JOUR AU - Balka, Richárd AU - Peres, Y TI - Uniform dimension results for fractional Brownian motion JF - JOURNAL OF FRACTAL GEOMETRY J2 - J FRACTAL GEOM VL - 4 PY - 2017 IS - 2 SP - 147 EP - 183 PG - 37 SN - 2308-1309 DO - 10.4171/JFG/48 UR - https://m2.mtmt.hu/api/publication/3338647 ID - 3338647 LA - English DB - MTMT ER - TY - JOUR AU - Balka, Richárd TI - Dimensions of fibers of generic continuous maps JF - MONATSHEFTE FUR MATHEMATIK J2 - MONATSH MATH VL - 184 PY - 2017 IS - 3 SP - 339 EP - 378 PG - 40 SN - 0026-9255 DO - 10.1007/s00605-017-1067-5 UR - https://m2.mtmt.hu/api/publication/3278973 ID - 3278973 N1 - First Online: 29 May 2017 November 2017 Export Date: 3 January 2019 Correspondence Address: Balka, R.; Department of Mathematics, University of British Columbia, and Pacific Institute for the Mathematical SciencesCanada; email: balka@math.ubc.ca Export Date: 10 January 2019 Correspondence Address: Balka, R.; Department of Mathematics, University of British Columbia, and Pacific Institute for the Mathematical SciencesCanada; email: balka@math.ubc.ca Export Date: 16 September 2019 Correspondence Address: Balka, R.; Department of Mathematics, University of British Columbia, and Pacific Institute for the Mathematical SciencesCanada; email: balka@math.ubc.ca Export Date: 30 September 2019 Correspondence Address: Balka, R.; Department of Mathematics, University of British Columbia, and Pacific Institute for the Mathematical SciencesCanada; email: balka@math.ubc.ca Export Date: 10 November 2020 Correspondence Address: Balka, R.; Department of Mathematics, University of British Columbia, and Pacific Institute for the Mathematical SciencesCanada; email: balka@math.ubc.ca AB - In an earlier paper Buczolich, Elekes, and the author described the Hausdorff dimension of the level sets of a generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space K by introducing the notion of topological Hausdorff dimension. Later on, the author extended the theory for maps from K to (Formula presented.). The main goal of this paper is to generalize the relevant results for topological and packing dimensions and to obtain new results for sufficiently homogeneous spaces K even in the case case of Hausdorff dimension. Let K be a compact metric space and let us denote by (Formula presented.) the set of continuous maps from K to (Formula presented.) endowed with the maximum norm. Let (Formula presented.) be one of the topological dimension (Formula presented.), the Hausdorff dimension (Formula presented.), or the packing dimension (Formula presented.). Define (Formula presented.)We prove that (Formula presented.) is the right notion to describe the dimensions of the fibers of a generic continuous map (Formula presented.). In particular, we show that (Formula presented.) provided that (Formula presented.), otherwise every fiber is finite. Proving the above theorem for packing dimension requires entirely new ideas. Moreover, we show that the supremum is attained on the left hand side of the above equation. Assume (Formula presented.). If K is sufficiently homogeneous, then we can say much more. For example, we prove that (Formula presented.) for a generic (Formula presented.) for all (Formula presented.) if and only if (Formula presented.) or (Formula presented.) for all open sets (Formula presented.). This is new even if (Formula presented.) and (Formula presented.). It is known that for a generic (Formula presented.) the interior of f(K) is not empty. We augment the above characterization by showing that (Formula presented.) for a generic (Formula presented.). In particular, almost every point of f(K) is an interior point. In order to obtain more precise results, we use the concept of generalized Hausdorff and packing measures, too. © 2017 Springer-Verlag Wien LA - English DB - MTMT ER - TY - JOUR AU - ANGEL, O AU - Balka, Richárd AU - PERES, Y TI - Increasing subsequences of random walks JF - MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY J2 - MATH PROC CAMBRIDGE VL - 163 PY - 2017 IS - 1 SP - 173 EP - 185 PG - 13 SN - 0305-0041 DO - 10.1017/S0305004116000797 UR - https://m2.mtmt.hu/api/publication/3122467 ID - 3122467 N1 - Published online: 23 September 2016 Megjegyzés-26865564 N1 Funding details: NSERC, Natural Sciences and Engineering Research Council of Canada N1 Funding text: Supported in part by NSERC. Supported by the National Research, Development and Innovation Office-NKFIH, 104178. Cited By :2 Export Date: 3 January 2019 Funding details: Natural Sciences and Engineering Research Council of Canada, NSERC Funding details: 104178 Funding text 1: Supported in part by NSERC. Supported by the National Research, Development and Innovation Office-NKFIH, 104178. Cited By :2 Export Date: 10 January 2019 Funding details: Natural Sciences and Engineering Research Council of Canada, NSERC Funding details: 104178 Funding text 1: Supported in part by NSERC. Supported by the National Research, Development and Innovation Office-NKFIH, 104178. Cited By :3 Export Date: 16 September 2019 Funding details: Natural Sciences and Engineering Research Council of Canada Funding details: 104178 Funding text 1: Supported in part by NSERC. Supported by the National Research, Development and Innovation Office-NKFIH, 104178. Cited By :7 Export Date: 10 November 2020 Funding details: Natural Sciences and Engineering Research Council of Canada, NSERC Funding details: 104178 Funding text 1: Supported in part by NSERC. Supported by the National Research, Development and Innovation Office-NKFIH, 104178. AB - Given a sequence of n real numbers {Si }i⩽n, we consider the longest weakly increasing subsequence, namely i 1 < i 2 < . . . < iL with Sik ⩽ Sik+1 and L maximal. When the elements Si are i.i.d. uniform random variables, Vershik and Kerov, and Logan and Shepp proved that (Formula presented.). We consider the case when {Si }i⩽n is a random walk on ℝ with increments of mean zero and finite (positive) variance. In this case, it is well known (e.g., using record times) that the length of the longest increasing subsequence satisfies (Formula presented.). Our main result is an upper bound (Formula presented.), establishing the leading asymptotic behavior. If {Si }i⩽n is a simple random walk on ℤ, we improve the lower bound by showing that (Formula presented.). We also show that if { S i } is a simple random walk in ℤ2, then there is a subsequence of { S i }i⩽n of expected length at least cn 1/3 that is increasing in each coordinate. The above one-dimensional result yields an upper bound of n 1/2+o(1). The problem of determining the correct exponent remains open. Copyright © Cambridge Philosophical Society 2016 LA - English DB - MTMT ER - TY - JOUR AU - Balka, Richárd AU - Darji, U B AU - Elekes, Márton TI - Bruckner–Garg-Type Results with Respect to Haar Null Sets in C[0,1] JF - PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY J2 - P EDINBURGH MATH SOC VL - 60 PY - 2017 IS - 1 SP - 17 EP - 30 PG - 14 SN - 0013-0915 DO - 10.1017/S0013091515000577 UR - https://m2.mtmt.hu/api/publication/3064913 ID - 3064913 N1 - Published online: 10 May 2016 Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, United States Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, PO Box 127, Budapest, 1364, Hungary Department of Mathematics, University of Louisville, Louisville, KY 40292, United States Eötvös Loránd University, Institute of Mathematics, Pázmány Péter s. 1/c, Budapest, 1117, Hungary Cited By :1 Export Date: 3 January 2019 Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, United States Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, PO Box 127, Budapest, 1364, Hungary Department of Mathematics, University of Louisville, Louisville, KY 40292, United States Eötvös Loránd University, Institute of Mathematics, Pázmány Péter s. 1/c, Budapest, 1117, Hungary Cited By :2 Export Date: 10 November 2020 Funding Agency and Grant Number: Hungarian Scientific Research FundOrszagos Tudomanyos Kutatasi Alapprogramok (OTKA) [72655, 104178, 83726] Funding text: R.B. was supported by the Hungarian Scientific Research Fund (Grants 72655 and 104178). U.B.D. would like to thank the Department of Analysis of Eotvos Lorand University for their hospitality. M.E. was supported by the Hungarian Scientific Research Fund (Grants 72655, 83726 and 104178). LA - English DB - MTMT ER -