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Let $b_1,\dots,b_n$ in $[-M,M]^m$ be linearly independent, and let $L = \sum_i \mathbb{Z}b_i$ be the lattice they generate. For $i = 1, \dots,n$, let $r_i$ be the smallest positive real number such th …

I assume you have checked this for small $n$, so I will only consider large $n$. I will show: if $n \geq 14$, then there exists a product $\sigma = (i_1 j_1) (i_2 j_2) \dots (i_5 j_5)$ of five transpo …

I doubt that there is an exact formula for this maximum, and unfortunately Wolfgang's guess is incorrect. Indeed, let $$ a_n = \mathrm{max}_{\sigma \in \mathfrak{S}_n} \sum_{i=1}^n \frac{i}{i + \sigma …

Let $R > 1$ and $\lambda \in \mathbb{R}$ be such that $$ \int_{1}^R \mathrm{tanh}(\frac{\lambda x}{2}) \frac{d x}{x} = \log R -2. $$ Then standard techniques in large deviation theory yield $$ \frac{ …

The simplest explanation is that it is a mistake. One can however complete the proof as follows: If $X$ is large enough then $u(m) \in \{ 0,1,2 \}$ for all $m$ since $e < 3$. Thus the number of disti …

Let $(S_i)_{i \in I}$ be a family of spheres of cardinality $|I| < \mathfrak{c}$ and let $S$ be a sphere distinct from each $S_i$. Each intersection $S \cap S_i$ is a (possibly degenerate) circle. Sin …

For fixed $d$, one can actually achieve a bound independent of $n$. More precisely, $K=K(d)=O(\sqrt{d})$ is fine, uniformly in $n$. Proof : the unit ball of $\mathbb{R}^d$ can be covered by $2C^d$ ba …

Yes. The coefficient of $x^n$ in $F_r$ is the $2n$-th derivative at $t = 0$ of the function $$ t \mapsto (\cos t)^{r} = \frac{1}{2^r} \sum_{k=0}^r \binom{r}{k} e^{it(2k-r)}. $$ But the successive deri …

The function $$ f : z \in \mathbb{C} \longmapsto \sum_{i} \frac{a_i}{1-a_iz} $$ is meromorphic on $\mathbb{C}$ and has integral Taylor coefficients. It follows from a theorem of Borel that such a fun …

There is an obvious bijective proof of the identity $$ 2 \binom{2n}{n} + 2 \binom{2n}{n + 1} = \binom{2n+2}{n+1}$$ and also a bijective proof of $$ 2\binom{2n}{n} - 2 \binom{2n}{n+1} = 2C_n,$$ see t …

Let $d_n$ be the maximal density of a squarefree set in $\mathbb F_2^n\oplus\mathbb F_2^n$. Then it is unknown whether there exists a constant $c < 1$ such that $d_n = O(c^n)$. Indeed, such a result w …

Let $0 \leq m \leq n$ be integers. The group $S_n$ of permutations acts on the ring $\mathbb{Z}[X_1,\dots,X_n]$ by permuting the coordinates, with fixed subring $\mathbb{Z}[\sigma_1,\dots,\sigma_n]$, …

I believe I have read or heard somewhere that the Kakeya conjecture would follow from appropriate lower bounds for the minimal size of a subset of $\{ 1 , \cdots , N\}$ which contains a translate of e …

Is there some minimal idempotent ultrafilter $q \in \beta( \mathbb{N}^2)$ (with respect to the law $"+"$) such that any $A \in q$ is a subset of $\mathbb{N} \times \{ 0 \} $ ? (See for example http: …

Let us compute the ordinary generating function of $k \mapsto c_{n+k,k}$, i.e. $S(X) = \sum_{k \geq 0 } c_{n+k,k} X^k $ (with notations as in Lierre's answer above) : $$ S(X^2) = \sum_j \sum_k \binom{ …