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The one-dimensional Rademacher differentiation theorem implies that the Cartesian product of two compact measure zero subsets of the real line is purely unrectifiable, which in turn can be used to establish the Besicovitch projection theorem, which asserts that if a subset of the plane has finite 1-dimensional Hausdorff measure and is purely unrectifiable, then almost every projection of that set to the real line has measure zero. Thus, for instance, if one takes the Cartesian product $C \times C$ of two copies of the middle-halves Cantor set $\{0,1\} \in C = \frac{1}{4} C \cup (\frac{1}{4} C + \frac{3}{4}) \subset [0,1]$, then almost every line in the plane will fail to hit this set. (Quantitative versions of this problem (commonly known as "Buffon's needle problem") have attracted attention in recent years, see e.g. the recent survey of Laba at http://arxiv.org/abs/1212.0247 .)

A few years ago with Hans Lindblad in http://arxiv.org/abs/1011.0949 , we used the one-dimensional Rademacher differentiation theorem to establish that solutions to a certain nonlinear wave equation in one spatial dimension necessarily decayed to zero as time went to infinity. This is in contrast to the linear wave equation which does not decay in one spatial dimension. The rough idea was that if the solution did not decay, then one could show that it concentrated along a Lipschitz curve in spacetime, which by Rademacher was approximately linear at some locations and some scales, and this could be shown to be in contradiction to a certain Morawetz-type estimate on solutions to nonlinear wave equations.

One fairly well known application of the higher-dimensional Rademacher differentiation theorem is by Pansu who extended this theorem to Carnot groups, and a variant of his theory establishes the fact that if two finitely generated nilpotent groups are quasiisometric, then their associated Carnot groups are isomorphic, which is still one of the strongest statements known about quasiisometry of groups in the nilpotent case.

The one-dimensional Rademacher differentiation theorem implies that the Cartesian product of two compact measure zero subsets of the real line is purely unrectifiable, which in turn can be used to establish the Besicovitch projection theorem, which asserts that if a subset of the plane has finite 1-dimensional Hausdorff measure and is purely unrectifiable, then almost every projection of that set to the real line has measure zero. Thus, for instance, if one takes the Cartesian product $C \times C$ of two copies of the middle-halves Cantor set $\{0,1\} \in C = \frac{1}{4} C \cup (\frac{1}{4} C + \frac{3}{4}) \subset [0,1]$, then almost every line in the plane will fail to hit this set. (Quantitative versions of this problem (commonly known as "Buffon's needle problem") have attracted attention in recent years, see e.g. the recent survey of Laba at https://arxiv.org/abs/1212.0247 .)

A few years ago with Hans Lindblad in https://arxiv.org/abs/1011.0949 , we used the one-dimensional Rademacher differentiation theorem to establish that solutions to a certain nonlinear wave equation in one spatial dimension necessarily decayed to zero as time went to infinity. This is in contrast to the linear wave equation which does not decay in one spatial dimension. The rough idea was that if the solution did not decay, then one could show that it concentrated along a Lipschitz curve in spacetime, which by Rademacher was approximately linear at some locations and some scales, and this could be shown to be in contradiction to a certain Morawetz-type estimate on solutions to nonlinear wave equations.

One fairly well known application of the higher-dimensional Rademacher differentiation theorem is by Pansu who extended this theorem to Carnot groups, and a variant of his theory establishes the fact that if two finitely generated nilpotent groups are quasiisometric, then their associated Carnot groups are isomorphic, which is still one of the strongest statements known about quasiisometry of groups in the nilpotent case.

The one-dimensional Rademacher differentiation theorem implies that the Cartesian product of two compact measure zero subsets of the real line is purely unrectifiable, which in turn can be used to establish the Besicovitch projection theorem, which asserts that if a subset of the plane has finite 1-dimensional Hausdorff measure and is purely unrectifiable, then almost every projection of that set to the real line has measure zero. Thus, for instance, if one takes the Cartesian product $C \times C$ of two copies of the middle-halves Cantor set $\{0,1\} \in C = \frac{1}{4} C \cup (\frac{1}{4} C + \frac{3}{4}) \subset [0,1]$, then almost every line in the plane will fail to hit this set. (Quantitative versions of this problem have attracted attention in recent years, see e.g. the recent survey of Laba at http://arxiv.org/abs/1212.0247 .)

A few years ago with Hans Lindblad in http://arxiv.org/abs/1011.0949 , we used the one-dimensional Rademacher differentiation theorem to establish that solutions to a certain nonlinear wave equation in one spatial dimension necessarily decayed to zero as time went to infinity. This is in contrast to the linear wave equation which does not decay in one spatial dimension. The rough idea was that if the solution did not decay, then one could show that it concentrated along a Lipschitz curve in spacetime, which by Rademacher was approximately linear at some locations and some scales, and this could be shown to be in contradiction to a certain Morawetz-type estimate on solutions to nonlinear wave equations.

One fairly well known application of the higher-dimensional Rademacher differentiation theorem is by Pansu who extended this theorem to Carnot groups, and a variant of his theory establishes the fact that if two finitely generated nilpotent groups are quasiisometric, then their associated Carnot groups are isomorphic, which is still one of the strongest statements known about quasiisometry of groups in the nilpotent case.

The one-dimensional Rademacher differentiation theorem implies that the Cartesian product of two compact measure zero subsets of the real line is purely unrectifiable, which in turn can be used to establish the Besicovitch projection theorem, which asserts that if a subset of the plane has finite 1-dimensional Hausdorff measure and is purely unrectifiable, then almost every projection of that set to the real line has measure zero. Thus, for instance, if one takes the Cartesian product $C \times C$ of two copies of the middle-halves Cantor set $\{0,1\} \in C = \frac{1}{4} C \cup (\frac{1}{4} C + \frac{3}{4}) \subset [0,1]$, then almost every line in the plane will fail to hit this set. (Quantitative versions of this problem (commonly known as "Buffon's needle problem") have attracted attention in recent years, see e.g. the recent survey of Laba at http://arxiv.org/abs/1212.0247 .)

A few years ago with Hans Lindblad in http://arxiv.org/abs/1011.0949 , we used the one-dimensional Rademacher differentiation theorem to establish that solutions to a certain nonlinear wave equation in one spatial dimension necessarily decayed to zero as time went to infinity. This is in contrast to the linear wave equation which does not decay in one spatial dimension. The rough idea was that if the solution did not decay, then one could show that it concentrated along a Lipschitz curve in spacetime, which by Rademacher was approximately linear at some locations and some scales, and this could be shown to be in contradiction to a certain Morawetz-type estimate on solutions to nonlinear wave equations.

One fairly well known application of the higher-dimensional Rademacher differentiation theorem is by Pansu who extended this theorem to Carnot groups, and a variant of his theory establishes the fact that if two finitely generated nilpotent groups are quasiisometric, then their associated Carnot groups are isomorphic, which is still one of the strongest statements known about quasiisometry of groups in the nilpotent case.

The one-dimensional Rademacher differentiation theorem implies that the Cartesian product of two compact measure zero subsets of the real line is purely unrectifiable, which in turn can be used to establish the Besicovitch projection theorem, which asserts that if a subset of the plane has finite 1-dimensional Hausdorff measure and is purely unrectifiable, then almost every projection of that set to the real line has measure zero. Thus, for instance, if one takes the Cartesian product $C \times C$ of two copies of the middle-halves Cantor set $\{0,1\} \in C = \frac{1}{4} C \cup (\frac{1}{4} C + \frac{3}{4}) \subset [0,1]$, then almost every line in the plane will fail to hit this set. (Quantitative versions of this problem have attracted attention in recent years, see e.g. the recent survey of Laba at http://arxiv.org/abs/1212.0247 .)

A few years ago with Hans Lindblad in http://arxiv.org/abs/1011.0949 , we used the one-dimensional Radamacher differentiation theorem to establish that solutions to a certain nonlinear wave equation in one spatial dimension necessarily decayed to zero as time went to infinity. This is in contrast to the linear wave equation which does not decay in one spatial dimension. The rough idea was that if the solution did not decay, then one could show that it concentrated along a Lipschitz curve in spacetime, which by Radamacher was approximately linear at some locations and some scales, and this could be shown to be in contradiction to a certain Morawetz-type estimate on solutions to nonlinear wave equations.

One fairly well known application of the higher-dimensional Rademacher differentiation theorem is by Pansu who extended this theorem to Carnot groups, and a variant of his theory establishes the fact that if two finitely generated nilpotent groups are quasiisometric, then their associated Carnot groups are isomorphic, which is still one of the strongest statements known about quasiisometry of groups in the nilpotent case.

The one-dimensional Rademacher differentiation theorem implies that the Cartesian product of two compact measure zero subsets of the real line is purely unrectifiable, which in turn can be used to establish the Besicovitch projection theorem, which asserts that if a subset of the plane has finite 1-dimensional Hausdorff measure and is purely unrectifiable, then almost every projection of that set to the real line has measure zero. Thus, for instance, if one takes the Cartesian product $C \times C$ of two copies of the middle-halves Cantor set $\{0,1\} \in C = \frac{1}{4} C \cup (\frac{1}{4} C + \frac{3}{4}) \subset [0,1]$, then almost every line in the plane will fail to hit this set. (Quantitative versions of this problem have attracted attention in recent years, see e.g. the recent survey of Laba at http://arxiv.org/abs/1212.0247 .)

A few years ago with Hans Lindblad in http://arxiv.org/abs/1011.0949 , we used the one-dimensional Rademacher differentiation theorem to establish that solutions to a certain nonlinear wave equation in one spatial dimension necessarily decayed to zero as time went to infinity. This is in contrast to the linear wave equation which does not decay in one spatial dimension. The rough idea was that if the solution did not decay, then one could show that it concentrated along a Lipschitz curve in spacetime, which by Rademacher was approximately linear at some locations and some scales, and this could be shown to be in contradiction to a certain Morawetz-type estimate on solutions to nonlinear wave equations.

One fairly well known application of the higher-dimensional Rademacher differentiation theorem is by Pansu who extended this theorem to Carnot groups, and a variant of his theory establishes the fact that if two finitely generated nilpotent groups are quasiisometric, then their associated Carnot groups are isomorphic, which is still one of the strongest statements known about quasiisometry of groups in the nilpotent case.