How large are the level sets of the Takagi function?

Let T be Takagi's continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire category. We first give more elementary proofs of three recently published results. The first, due to Z. Buczolich, states that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states that the average number of points in a level set is infinite. The third result, also due to Lagarias and Maddock, states that the average number of local level sets contained in a level set is 3/2. In the second part of the paper it is shown that, in contrast to the above results, the set of ordinates y with uncountably infinite level sets is residual, and a fairly explicit description of this set is given. The paper also gives a negative answer to a question of Lagarias and Maddock by showing that most level sets (in the sense of Baire category) contain infinitely many local level sets, and that a continuum of level sets even contain uncountably many local level sets. Finally, several of the main results are extended to a version of T with arbitrary signs in the summands.Comment: Added a new Section 5 with generalization of the main results; some new and corrected proofs of the old material; 29 pages, 3 figure

Spitzer Observations of the Predicted Eddington Flare from Blazar OJ 287

Binary black hole (BH) central engine description for the unique blazar OJ 287 predicted that the next secondary BH impact-induced bremsstrahlung flare should peak on 2019 July 31. This prediction was based on detailed general relativistic modeling of the secondary BH trajectory around the primary BH and its accretion disk. The expected flare was termed the Eddington flare to commemorate the centennial celebrations of now-famous solar eclipse observations to test general relativity by Sir Arthur Eddington. We analyze the multi-epoch Spitzer observations of the expected flare between 2019 July 31 and 2019 September 6, as well as baseline observations during 2019 February-March. Observed Spitzer flux density variations during the predicted outburst time display a strong similarity with the observed optical pericenter flare from OJ 287 during 2007 September. The predicted flare appears comparable to the 2007 flare after subtracting the expected higher base-level Spitzer flux densities at 3.55 and 4.49 μm compared to the optical R-band. Comparing the 2019 and 2007 outburst lightcurves and the previously calculated predictions, we find that the Eddington flare arrived within 4 hr of the predicted time. Our Spitzer observations are well consistent with the presence of a nano-Hertz gravitational-wave emitting spinning massive binary BH that inspirals along a general relativistic eccentric orbit in OJ 287. These multi-epoch Spitzer observations provide a parametric constraint on the celebrated BH no-hair theorem. © 2020. The American Astronomical Society. All rights reserved.

Spitzer Observations of the Predicted Eddington Flare from Blazar OJ 287

© 2020. The American Astronomical Society. All rights reserved.. Binary black hole (BH) central engine description for the unique blazar OJ 287 predicted that the next secondary BH impact-induced bremsstrahlung flare should peak on 2019 July 31. This prediction was based on detailed general relativistic modeling of the secondary BH trajectory around the primary BH and its accretion disk. The expected flare was termed the Eddington flare to commemorate the centennial celebrations of now-famous solar eclipse observations to test general relativity by Sir Arthur Eddington. We analyze the multi-epoch Spitzer observations of the expected flare between 2019 July 31 and 2019 September 6, as well as baseline observations during 2019 February-March. Observed Spitzer flux density variations during the predicted outburst time display a strong similarity with the observed optical pericenter flare from OJ 287 during 2007 September. The predicted flare appears comparable to the 2007 flare after subtracting the expected higher base-level Spitzer flux densities at 3.55 and 4.49 μm compared to the optical R-band. Comparing the 2019 and 2007 outburst lightcurves and the previously calculated predictions, we find that the Eddington flare arrived within 4 hr of the predicted time. Our Spitzer observations are well consistent with the presence of a nano-Hertz gravitational-wave emitting spinning massive binary BH that inspirals along a general relativistic eccentric orbit in OJ 287. These multi-epoch Spitzer observations provide a parametric constraint on the celebrated BH no-hair theorem