Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Erdos-Littlewood-Offord problem with arbitrary probabilities   The classical Erd\H{o}s-Littlewood-Offord problem concerns the random
variable $X = a_1 \xi_1 + \dots + a_n \xi_n$, where $a_i \in \mathbb{R}
\setminus \{0\}$ are fixed and $\xi_i \sim \text{Ber}(1/2)$ are independent.
The Erd\H{o}s-Littlewood-Offord theorem states that the maximum possible
concentration probability $\max_{x \in \mathbb{R}} \Pr(X = x)$ is
$\binom{n}{\lfloor n/2\rfloor} / 2^n$, achieved when the $a_i$ are all $1$. As
proposed by Fox, Kwan, and Sauermann, we investigate the general case where
$\xi_i \sim \text{Ber}(p)$ instead. Using purely combinatorial techniques, we
show that the exact maximum concentration probability is achieved when $a_i \in
\{-1, 1\}$ for each $i$. Then, using Fourier-analytic techniques, we
investigate the optimal ratio of $1$s to $-1$s. Surprisingly, we find that in
some cases, the numbers of $1$s and $-1$s can be far from equal.

32
The categories of this paper are: atom-ph
19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
37
The categories of this paper are: supr-con
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ