Find, on a sequence of size L, the number that appears more than L/2 times
Bug / unexpected result

Focusing only on mostFrequentElement() method  

Assume passing the array {2,2,3,3,3} this will result in "None" but the correct result should be "3".  

To get rid of this unwanted result, we need after the loop to increment the counter by 1 and add a check   

counter++;
if (counter &gt; bigCounter) {
    big = arr[arr.length - 1];
    bigCounter = counter;
}


Simplification  

Because you are incrementing counter wether the if condition is true or false this can be simplified to  

for (int i = 0; i &lt; arr.length - 1; i++) {
    counter++;
    if (arr[i] != arr[i + 1]) {
        if (counter &gt; bigCounter) {
            big = arr[i];
            bigCounter = counter;
        }                        
        counter = 0;
    }            
}


General  


Instead of return big + ""; you should use the return String.valueOf(big); because it is more readable.  
you should also check if the given array contains any element.  
declaring multiple variables on one line removes readability.  
you can initialize bigCounter with 0 because counter is always &gt; 0.  
big and bigCounter aren't good names. Better name them mostFrequentElement and mostFrequentElementCounter and rename the method to getMostFrequentElement.
Using verbs or verb phrases for methodnames improves readability.  


Refactoring  

private static String getMostFrequentElement(int[] arr) {
    if (arr.length == 0) {
        return "None";
    }

    Arrays.sort(arr);

    int counter = 0;
    int mostFrequentElement = -1;
    int mostFrequentElementCounter = 0;

    for (int i = 0; i &lt; arr.length - 1; i++) {
        counter++;
        if (arr[i] != arr[i + 1]) {
            if (counter &gt; mostFrequentElementCounter) {
                mostFrequentElement = arr[i];
                mostFrequentElementCounter = counter;
            }
            counter = 0;
        }
    }

    counter++;
    if (counter &gt; mostFrequentElementCounter) {
        mostFrequentElement = arr[arr.length - 1];
        mostFrequentElementCounter = counter;
    }

    if (mostFrequentElementCounter &gt; arr.length / 2) {
        return String.valueOf(mostFrequentElement);
    }
    return "None";
}