A note on the dimensions of difference and distance sets for graphs of functions

Manuj Verma and Amit Priyadarshi

Chaos, Solitons & Fractals, 2024, vol. 184, issue C

Abstract: In this article, we discuss the Hausdorff dimensions of the distance sets and the difference sets for the graphs of continuous functions on the unit interval. We also prove that the distance set conjecture is true for a dense subset of (B([0,1]),‖⋅‖p). After that, we determine bounds on the dimension of the difference and distance sets for the graph of a function on an uncountable bounded domain in terms of the dimension of the graph of the function. Lastly, we determine a non-trivial lower bound for the upper box dimension of the difference set of a set in the plane and also discuss the dimension of the distance set of the product of sets.

Keywords: Distance set; Hausdorff dimension; Box dimension; Graphs of functions; Continuous functions; Bounded functions (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:184:y:2024:i:c:s0960077924005381

DOI: 10.1016/j.chaos.2024.114986

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