Let $\ell \ne 3$ be a prime. We show that there are only finitely many cyclic number fields $F$ of degree $\ell$ for which the unit equation $$\lambda + \mu = 1, \qquad \lambda,~\mu \in \mathcal{O}_F^\times$$ has solutions. Our result is effective. For example, we deduce quintic field solutions $\mathbb{Q}(\zeta_{11})^+$.

The distribution of $$\alpha p$$ modulo one, where p runs over the rational primes and $$ is a fixed irrational real, has received lot attention. It natural to ask for which exponents $$\nu >0$$ one can establish infinitude satisfying $$||\alpha p||\le p^{-\nu }$$ . latest record in this regard Kaisa Matomäki’s landmark result =1/3-\varepsilon presents limit currently known technology. Recently...

We show that every Dedekind domain $R$ lying between the polynomial rings $\mathbb Z[X]$ and Q[X]$ with property its residue fields of prime characteristic are finite is equal to a generalized ring integer-valued polynomials, is, for each $p\in\mathbb Z$ there exists subset $E_p$ transcendental elements over Q$ in absolute integral closure $\overline{\mathbb Z_p}$ $p$-adic integers such $R=\{f\...

Higher-order Fourier analysis, developed over prime fields, has been recently used in different areas of computer science, including list decoding, algorithmic decomposition and testing. We extend the tools of higher-order Fourier analysis to analyze functions over general fields. Using these new tools, we revisit the results in the above areas. (i) For any fixed finite field K, we show that th...