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Finding arithmetic progressions in dense sets of integers
Sarah Peluse
https://arxiv.org/abs/2509.22962 https://arxiv.org/pdf/2509.22962

Finding arithmetic progressions in dense sets of integers
One of the central problems in additive combinatorics is to determine how large a subset of the first $N$ integers can be before it is forced to contain $k$ elements forming an arithmetic progression. Around 25 years ago, Gowers proved the first reasonable upper bounds in this problem for progressions of length four and longer. In this work, Gowers initiated the study of "higher-order Fourier analysis", which has developed over the past couple of decades into a rich theory with numerous other c…

Mercury Crustal Magnetization Indicates a Stronger Ancient Dynamo
Isaac S. Narrett, Benjamin P. Weiss, Sarah C. Steele, John B. Biersteker
https://arxiv.org/abs/2510.02003 https…

Mercury Crustal Magnetization Indicates a Stronger Ancient Dynamo
Mercury is the only other terrestrial planet in the solar system with an active dynamo magnetic field (~200 nT at the equatorial surface). Furthermore, Mercury's ~3.9-3.7 billion-year old (Ga) crust is strongly magnetized (~10 nT at ~30-km altitude), indicating the presence of a past dynamo. However, the source and strength of the field that magnetized this crust are unknown. To address this, we performed three-dimensional magnetohydrodynamic simulations of the ancient solar wind interaction wi…