Fourier Dimension and Translation Invariant Linear Equations
Angel D. Cruz
TL;DR
This paper proves that any subset of the real line with Fourier dimension exceeding 1/2 necessarily contains solutions to certain translation invariant linear equations involving four variables.
Contribution
It establishes a new threshold for Fourier dimension ensuring the existence of solutions to specific linear equations, advancing understanding in harmonic analysis and additive combinatorics.
Findings
Sets with Fourier dimension > 1/2 contain solutions to the equation
Provides a link between Fourier dimension and solutions to linear equations
Advances the theory connecting harmonic analysis and additive structures
Abstract
We consider a translation invariant linear equation in four variables with integer coefficients of the form: . The main result of the paper states that any set on the real line with Fourier dimension greater than 1/2 must contain a nontrivial solution of such an equation.
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Taxonomy
TopicsMatrix Theory and Algorithms · Numerical methods for differential equations · Elasticity and Wave Propagation