Project Details
Description
Value: 217950 Eur
This project lies at the intersection between Diophantine and metric number theory. The starting point for the first part of the project is Pillai’s asymptotic formula for counting differences between two powers with fixed integers bases. We can easily replace the integer bases by algebraic numbers. However, when we move on to general real bases, this the associated problem naturally becomes a metric problem one with many open questions. The second part of the project focuses on the shrinking target problem for matrix transformations on the d-dimensional torus. This was recently solved in terms of Lebesgue-measure and Hausdorff-dimension. However, the problem restricted to manifolds is open, and some partial results are conditional on the validity of the abc-conjecture. The last part of the project is concerned with lacunary sums. They are well understood in the classical regimes of the CLT and LIL. However, deviations at larger scales depend on Diophantine effects are little is known.
This project lies at the intersection between Diophantine and metric number theory. The starting point for the first part of the project is Pillai’s asymptotic formula for counting differences between two powers with fixed integers bases. We can easily replace the integer bases by algebraic numbers. However, when we move on to general real bases, this the associated problem naturally becomes a metric problem one with many open questions. The second part of the project focuses on the shrinking target problem for matrix transformations on the d-dimensional torus. This was recently solved in terms of Lebesgue-measure and Hausdorff-dimension. However, the problem restricted to manifolds is open, and some partial results are conditional on the validity of the abc-conjecture. The last part of the project is concerned with lacunary sums. They are well understood in the classical regimes of the CLT and LIL. However, deviations at larger scales depend on Diophantine effects are little is known.
Status | Not started |
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Effective start/end date | 1/09/25 → 31/08/27 |