We prove that every recovering pair $(\mathcal{A},\mathcal{B})$ of nonempty families of subsets of an $n$-element set satisfies $$|\mathcal{A}||\mathcal{B}|\le\left(\frac{223949}{100000}\right)^n=(2.23949)^n.$$ A result of Mond, Souza and Versteegen, combined with the sharp $(9/4)^n$ bound of Fang and Huang for cancellative pairs, gives $(2.2499)^n$ for recovering pairs. We improve this to $(2.23949)^n$. The proof combines two entropy inequalities. One is derived directly from the recovering property, while the other applies the Fang-Huang bound to families obtained from the original pair by a construction of Mond, Souza and Versteegen. The two inequalities are then combined by a convex combination.
We establish two quantitative results concerning additive decompositions of the set $\mathbb{P}$ of primes. First, we prove that, if $A,B\subseteq\mathbb N$ have at least two elements and $A+B$ differs from the primes by only finitely many elements, then $$ \frac{\sqrt{x}}{\log x}\ll A(x),B(x) \ll \sqrt{x}. $$ Second,...
For every finite $P\subset\mathbb{R}^2$ we prove $|\{p\cdot q: p,q\in P\}|\gg |P|^{199/295}$, with an absolute constant and no logarithmic loss, where $199/295 = 2/3+7/885$. This improves the bound $2/3+7/1425$ of Kokkinos (arXiv:2502.12727), which in turn had improved the first superthreshold bound of Hanson, Roche-Ne...
Fuglede's conjecture in $\mathbb{F}_p^d$ says that a nonempty set tiles by translation if and only if it is spectral. We prove the conjecture in $\mathbb{F}_2^8$ and $\mathbb{F}_{11}^3$. Over $\mathbb{F}_2$ we show more generally that the Fourier support of a nonconstant indicator of density different from one half con...
We prove that every set $\mathcal{B}\subseteq\mathbb{Z}_{\ge0}$ for which $\mathbb{P}+\mathcal{B}$ contains all sufficiently large integers satisfies $B(X)\gg\log X\log\log X$, where $B(X)=\#(\mathcal{B}\cap[0,X])$ and the implied constant is absolute. This proves a quantitative form of Ruzsa's Conjecture 2 and gives a...
For every negative fundamental discriminant $-f$ and every $k\geq1$, we construct a rational function $R_{f,2k}\in\mathbb{Q}(x_1,\ldots,x_{2k})$ and a constant $r_{f,2k}\in\mathbb{Q}^\times$ such that \[ m(R_{f,2k})=r_{f,2k}L'(\chi_{-f},1-2k), \] where $m$ denotes the logarithmic Mahler measure and $\chi_{-f}$ is the q...
We prove that a nonempty subset of the binary vector space $\mathbb{F}_2^7$ is spectral if and only if it tiles $\mathbb{F}_2^7$ by translations. This is the smallest binary dimension left open by Ferguson and Sothanaphan. For the implication from tiling to spectrality we show that every eight-point subset of $\mathbb{...
Atsushi Kadowaki· 0 citations
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