Description
This talk is exactly what the title promises: a medley of three distinct topics in number theory and additive combinatorics.
First, in joint work with Imre Ruzsa, and inspired by a beautiful proof by Béla Brindza, we discuss an ABC-type theorem for polynomials over $\mathbb{C}$ or $\overline{\mathbb{F}_p}$. We establish degree bounds for polynomials satisfying $\sum_{i=1}^m f_i = 0$ under certain coprimality conditions. This leads to neat corollaries bounding the exponent $k$ in equations like $\sum_{i=1}^m \lambda_i p_i^k = 0$ or
$\sum_{i=1}^m \lambda_i p_i^k = P(t)$, which yield further applications in the theory of pseudorandomness.
Second, while wrapping up the aforementioned research, Imre shared one of his favorite open problems regarding solution-free sets for asymmetric linear equations. I spent a lot of time thinking about $2X+2Y = U+3Z$ and essentially got nowhere. Fortunately, the equation $X+5Y = 3U+3Z$ proved to be much more forgiving! I will show explicit constructions for the latter that successfully break the expected $0.5$ logarithmic density barrier. As it turns out, József Solymosi and I were scratching our heads over very similar questions at the exact same time, so I will briefly mention this fun parallel.
Finally, since my original plan for this seminar was to talk about prime gaps, I will sneak in a quick result at the end. I will present a theorem showing that there exist infinitely many integers $n$ for which the gap $p_{n+1}-p_n < e^{1.834 \cdot 10^{76}}$ and is a perfect power.
Note: The slides will be in English, but the presentation will be delivered in Hungarian.
Zoom: https://us06web.zoom.us/j/88299143443?pwd=v0aQr1RuUANmVALlszcdAIwBSrzcYq.1
Meeting ID: 882 9914 3443
Passcode: 494643