Let $s_2(n)$ be the binary sum-of-digits function and let $c_t$ be the natural density of the integers $n\ge0$ for which $s_2(n+t)\ge s_2(n)$. Earlier work of the author proved the universal exponential bound $$c_t-\frac12\ge 2^{-2s_2(t)-1},$$ thereby resolving Cusick's conjecture for every $t$. This estimate, however, does not reflect the true size of the smallest possible bias at a given large Hamming weight. In this paper, we determine this extremal scale sharply: $$\inf_{s_2(t)=k}\left(c_t-\frac12\right) \sim \frac{1}{2\sqrt\pi} \left(\frac{\log_2 k}{k}\right)^{3/2} \qquad(k\to\infty).$$ Thus the optimal fixed-weight gap is polynomial-logarithmic rather than exponential, with the explicit sharp leading constant $1/(2\sqrt\pi)$. The proof combines the five-cumulant Edgeworth expansion of Spiegelhofer and Wallner with a new extremal rigidity mechanism for near-extremal binary block patterns. We also prove a stability theorem for asymptotic extremizers and give a separate shadow-energy interpretation of the same constant.
In 2019 La Bret\`eche, Stoll and Tennenbaum showed that the ratio of the sum of digits function $s_{q_1}(n)/s_{q_2}(n)$ of two multiplicatively independent bases $q_1$ and $q_2$ is dense in $\mathbb{Q}^+$. Recently Spiegelhofer proved that in the special case $s_2(n)/s_3(n)=1$ we have infinitely many solutions. Spiegel...
Let $s(n)$ denote the number of ones in the binary expansion of an integer $n\in\mathbb{N}$, and let $\mu_t$ be the probability measure on $\mathbb{Z}$ defined by the asymptotic densities of the level sets of the function $\mathbb{N}\ni n\mapsto s(n+t)-s(n)\in\mathbb{Z}$. Let $P_t$ be the family of finitely supported m...
We study several probabilistic questions concerning the digits of $s(n)$, the sum of proper divisors of an integer $n$. In particular, we show that $s(n)$ obeys Benford's law with respect to logarithmic density. Moreover, we show that, for every function $k(x) \rightarrow \infty$, almost all integers $n \leq x$ have ev...
Kubra Benl.i, Cécile Dartyge, Charlotte Dombrowsky et al.· 0 citations
Let $g_n$ be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in $\mathbb{R}^n$. We study this problem for finite sets invariant under all coordinate permutations. We prove that the exponential growth rate in this symmetric problem can be characterized exactly as...
Andrii Arman, A. Bondarenko, A. Prymak et al.· 0 citations
Let $P^+(n)$ be the largest prime factor of $n$, and let $\chi=\chi_{-4}$ be $1$ on primes $1\pmod4$ and $-1$ on primes $3\pmod4$. For fixed $k\ge2$ we study \[ D_k(x)=\sum_{\substack{n\le x\\ \Omega(n)=k}}\chi(P^+(n)). \] Thus $D_k(x)$ compares the two residue classes according to the largest prime factor of integers...
Let $S_k(\alpha;K)$ denote the exponential sum over $k$-free integers in the short interval $(N-K,N]$. For $s>0$, we prove essentially tight bounds on the $s$-th moments of $S_k(\alpha;K)$ whenever $K \gg N^{\theta_{k,s}+\epsilon}$ for some $\theta_{k,s}<1/2$. As an immediate consequence, we obtain a lower bound for th...
B. Doyle· 0 citations
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